In this paper, the control problems of linear dynamic systems stage by stage changing and the optimal control with the criteria of quality set for the whole range of time intervals are considered. The necessary and sufficient conditions of total controllability are also stated. The constructive solving method of a control problem is offered, as well as the definitions of conditions for the existence of programmed control and motions. The explicit form of control action for a control problem is constructed.
Yugoslav Journal of Operations Research 22 (2012), Number 1, 31-39 DOI:10.2298/YJOR111019002B CONTROL OF STAGE BY STAGE CHANGING LINEAR DYNAMIC SYSTEMS V.R BARSEGHYAN Yerevan State University Armenia, Yerevan barseghyan@sci.am Received: October 2011 / Accepted: November 2011 Abstract: In this paper, the control problems of linear dynamic systems stage by stage changing and the optimal control with the criteria of quality set for the whole range of time intervals are considered The necessary and sufficient conditions of total controllability are also stated The constructive solving method of a control problem is offered, as well as the definitions of conditions for the existence of programmed control and motions The explicit form of control action for a control problem is constructed The method for solving optimal control problem is offered, and the solution of optimal control of a specific target is brought Keywords: Control problem, optimal control, dynamic system, stage by stage changing MSC: 37N35, 93C05, 49J15, 93C15 INTRODUCTION Problems of control, and an optimal control of objects on the set of operational states have an important role, theoretical and applied, for instance, in the case of the change of dynamic model’s parameters due to the operation of system These problems are typical for the energy-efficient and thermal devices, the machinery with electricity cables and other mechanisms of industrial-technological purpose In particular, the dynamic model of controllable object with the change of operational state confined, when time intervals are strictly limited, is considered in [1, 2] Both in the ordinary control problems and the problems of dynamic systems changing stage by stage, the challenges of phased controllability, control, and optimal control emerge [3-6] The control and optimal control problems of linear dynamic V R Barseghyan / Control of Stage by Stage 32 systems and stage by stage changing linear dynamic systems are stated and investigated in [2] and [7] The control and the optimal control problems of linear dynamic systems changing stage by stage are stated and investigated in this paper PROBLEMS' DEFINITIONS Consider a control process, dynamics of which is described by the phased changing linear differential equations ⎧ A1 (t ) x + B1 (t )u , for t ∈ [t0 , t1 ) ⎪ ⎪ A (t ) x + B2 (t )u, for t ∈ [t1 , t2 ) &x = ⎨ M ⎪ ⎪⎩ Am (t ) x + Bm (t )u , for t ∈ [tm −1 , T ] (2.1) where x(t ) ∈ R n is the phase vector of system, Ak (t ) , Bk (t ) (k = 1, …, m) are matrices (n × n) and (n × r ) of parameters of the system (the models of the object) respectively, u (t ) is column-vector (r × 1) of control action In general, we will assume that the entries of matrices of functions Ak (t ) , Bk (t ) and the entries of column-vector u (t ) are measurable bounded functions There is given initial x(t0 ) = x0 (2.2) x(T ) = xT (2.3) and final states of the system (2.1) It is assumed that at given interim moments of time ≤ t0 < t1 < < tm −1 < tm = T the end of previous period motion is the beginning of the next period motion, i.e at t k moments of time x(tk − 0) = x(tk + 0) = x(tk ) , for (k = 1, , m − 1) Consider the following problems Problem Find conditions of the existence of programmed control action u = u (t ) transferring the motion of system (2.1) from the initial state (2.2) to the final state (2.3) at the time interval [t0 , T ] , and construct them as well For the selection of optimal solutions at the time interval [t0 , T ] , the criteria of quality æ [u ] which can have a sense of norm for some normed space, is given For the system (2.1) with states (2.2), (2.3) and the criteria of quality æ [u ] , the problem of optimal control can be defined as follows V R Barseghyan / Control of Stage by Stage 33 Problem Find the optimal control action by moving the system (2.1) from the initial state(2.2) to the final state (2.3) and having the lowest possible value of criteria of quality æ ⎡⎣u ⎤⎦ SOLUTION OF PROBLEMS To solve the defined problems, write the solution of system (2.1) for the time interval [tk −1 , tk ] as follows: t x(t ) = X k [t , tk −1 ]x(tk −1 ) + ∫H k [t ,τ ]u (τ )dτ , (3.1) tk −1 where H k [t ,τ ] = X k [t ,τ ]Bk (τ ) and X k [t ,τ ] is the normed fundamental matrix of solution of the homogeneous part of equation k of the system (2.1) for a time interval [tk −1 , tk ] x& = Ak (t ) x + Bk (t )u Assuming that the required control actions for t = tk are known, from (3.1) for t = tk moments of time, we have x(tk ) = X k [tk , tk −1 ]x(tk −1 ) + tk ∫ H k [tk ,τ ]u(τ )dτ (3.2) tk −1 Substituting the expression for x(tk −1 ) into (3.1), obtained from writing (3.2) for the previous period [tk − , tk −1 ] , we get x(t ) = X k [t , tk −1 ] X k −1[tk −1 , tk − ]x(tk − ) + tk −1 t tk − tk −1 + X k [t , tk −1 ] ∫ H k −1[tk −1 ,τ ]u (τ )dτ + ∫ H k [t ,τ ]u (τ )dτ (3.3) If we continue this process for the previous time intervals, we will get the formula, describing the motion of system (2.1) for t ∈ [tk −1 , tk ] moment of time k −1 tj j =1 t j −1 x(t ) = V (t , t0 ) x(t0 ) + ∑ V (t , t j ) ∫ H j [t j ,τ ]u (τ )dτ + (3.4) t + ∫ H k [t ,τ ]u (τ )dτ tk −1 where the following notations are used k − j −1 V (t , t j ) = X k [t , tk −1 ]V (tk −1 , t j ), V (tk , t j ) = ∏ X k −i [tk −i , tk −i −1 ], i =0 (k = 1, , m; j = 0, , k − 1) (3.5) V R Barseghyan / Control of Stage by Stage 34 According to the introduced notations, when j = k − , V (tk , tk −1 ) = X k [tk , tk −1 ] When j = k , V (tk , tk ) = E , then for t = tk (3.4) can be written in the following form k tj j =1 t j −1 m tj j =1 t j −1 x(tk ) = V (tk , t0 ) x(t0 ) + ∑ V (tk , t j ) ∫ H j [t j , t ]u (t )dt for k = m x(Т ) = V (Т , t0 ) x(t0 ) + ∑ V (Т , t j ) ∫ H j [t j , t ]u (t )dt (3.6) So, having an initial state x(t0 ) of system (2.1) and giving the control action u (t ) , the phase state x(t ) of system (2.1) can be determined for any point of time t from [tk −1 , tk ) using equation (3.4) Now instead of H k [tk , t ] , we introduce H k [tk , t ] function in the following form ⎧ H1[t1 , t ], for t0 ≤ t < t1 H1[t1 , t ] = ⎨ ⎩0, for t1 ≤ t ≤ T ⎧0, for t0 ≤ t < tk −1 ⎪ H k [tk , t ] = ⎨ H k [tk , t ], for tk −1 ≤ t < tk ⎪ ⎩0, for tk ≤ t ≤ T (3.7) k = 2, , m − ⎧0, for t0 ≤ t < tm −1 H m [tm , t ] = ⎨ ⎩ H m [tm , t ], for tm −1 ≤ t ≤ T Using functions introduced in (3.7), the relation (3.6) can be rewritten as T m ∫ ∑V (Т , t t0 j =1 j ) H j [t j , t ]u (t )dt = x(Т ) − V (Т , t0 ) x(t0 ) = C (3.8) Note, that there are n integral relations in (3.8) Thereby, the system (2.1) is totally controllable if and only if for any C = x(Т ) − V (Т , t0 ) x (t0 ) vector from R n There can be found control u = u (t , C ) satisfying condition (3.8) The controlability analysis of any system is important for the solution of control problem ′ ⎛ m ⎞ Let, hi (T , t ) be the i -th column of matrix ⎜⎜ ∑ V (Т , t j ) H j [t j , t ] ⎟⎟ ; Ci be the ⎝ j =1 ⎠ i th component of vector C Here and further, prime denotes an operation of transposition Then equation (3.8) can be written V R Barseghyan / Control of Stage by Stage 35 T ∫ h′(T , t )u (t )dt = C i i (i = 1,K , n) t0 So, the condition of controllability of system (2.1) can be formulated in the following theorem Theorem The system (2.1) is totally controllable in the interval [t0 , T ] if and only if the vector-functions h1 (T , t ),L , hn (T , t ) are linearly independent in that interval Now, the function u (t ) , satisfying the integral relation (3.8), has the form [5] ′ ⎛ m ⎞ u (t ) = ⎜⎜ ∑ V (Т , t j ) H j [t j , t ] ⎟⎟ η + v(t ) , ⎝ j =1 ⎠ (3.9) where η - constant vector (will be defined further), v(t ) -some vector function (can be a measurable bounded function in the time interval [t0 , T ] ), for which ⎛ m ⎞ V (Т , t j ) H j [t j , t ] ⎟⎟v(t )dt = ∫t ⎜⎜⎝ ∑ j =1 ⎠ T (3.10) Equation (3.10) is the condition of orthogonality of a vector-function v ( t ) to ⎛ m ⎞ all rowes of matrix ⎜⎜ ∑ V (Т , t j ) H j [t j , t ] ⎟⎟ ⎝ j =1 ⎠ Substituting (3.9) into (3.8), we get Q(t0 , , T )η = C , (3.11) ′ T ⎛ m ⎞⎛ m ⎞ Q(t0 , , T ) = ∫ ⎜⎜ ∑ V (Т , t j ) H j [t j , t ] ⎟⎟ ⎜⎜ ∑ V (Т , t j ) H j [t j , t ] ⎟⎟ dt ⎠ ⎝ j =1 ⎠ t0 ⎝ j =1 (3.12) where (3.11) is a system of n algebraic equations with C j ( j = 1,K , n) unknowns Equation (3.11) can be solved if either det Q ≠ or the matrix Q , and extended matrix {Q, C} have the same rank The solution of (3.11) will be of this form η = Q −1C hence, taking into account the value of vector C , (3.9) is written as ′ ⎛ m ⎞ −1 u (t ) = ⎜⎜ ∑ V (Т , t j ) H j [t j , t ] ⎟⎟ Q ( x(Т ) − V (Т , t0 ) x(t0 )) + v(t ) ⎝ j =1 ⎠ (3.13) V R Barseghyan / Control of Stage by Stage 36 So, the solution of Problem can be formulated in the following theorem Theorem Program control (3.13) and the corresponding solution of system (2.1), satisfying (2.2) and (2.3), exists if and only if, the matrix (3.12) isn’t singular, or matrices Q and {Q, C} have the same rank Taking into account notations in (3.7), the control action u (t ) when v ( t ) = according to (3.13), is written in the following form ⎧(V (T , t1 ) H1[t1 , t ])′Q −1 ( x(T ) − V (T , t0 ) x(t0 )), for t ∈ [t0 , t1 ) ⎪ −1 ⎪(V (T , t2 ) H [t2 , t ])′Q ( x(T ) − V (T , t0 ) x(t0 )), for t ∈ [t1 , t2 ) u (t ) = ⎨ ⎪ −1 ⎪ ⎩( H m [tm , t ])′Q ( x(T ) − V (T , t0 ) x(t0 )), for t ∈ [tm −1 , T ] (3.14) Substituting (3.14) into (3.4), and, taking into account the continuity of motion in the intermediate moments of time tk , we get the program motion of system (2.1) in [t0 , T ] , and satisfying conditions (2.2), (2.3) For the solution of Problem 2, note the following For the given quality criterion æ [u ] , the problem of optimal control with the integral conditions (3.8) can be considered as a problem of a conditional extremum from the variation calculus, where one should find minimum of functional æ [u ] subject to (3.8) However, as it is obvious from (3.7), the sub integral functions in (3.8) are disruptive and because of it, the theorems of variation calculus aren't valid for solving this problem [4] The left hand side of condition (3.8) is linear operation generated by the function u (t ) in [t0 , T ] time interval Hence, if functional æ [u ] is a norm of some linear normed space, the solution of Problem should be find using the problem of moments, then, the optimal control action u (t ) , t ∈ [t0 , T ] minimizing functional æ [u ] is the solution to Problem So, the problem is reduced to the problem of moments, whose solution is known from [4] EXAMPLE As an illustration of the presented method for constructing optimal control action consider the model of the controled object, described in [2], whose dynamics of changes stage by stage has the form ⎧⎪ x&1 = x2 ⎨ (1) ⎪⎩ x&2 = b u ⎧⎪ x&1 = x2 ⎨ (2) (2) ⎪⎩ x&2 = a2 x2 + b u t ∈ [t0 , t1 ) (4.1) t ∈ [t1 , t2 ) (4.2) V R Barseghyan / Control of Stage by Stage ⎧⎪ x&1 = x2 ⎨ (3) (3) (3) ⎪⎩ x&2 = a1 x1 + a2 x2 + b u t ∈ [t2 , t3 ] 37 (4.3) where a2(2) , a1(3) , a2(3) , b(1) , b(2) , b(3) are the parameters of object model and t0 < t1 < t2 < t3 are the intermediate moments of time Initial and final phase states are given as ⎛ x1 (t0 ) ⎞ ⎛ x1 (t3 ) ⎞ x(t0 ) = ⎜ ⎟ , x(t3 ) = ⎜ ⎟ ⎝ x2 (t0 ) ⎠ ⎝ x2 (t3 ) ⎠ A quality criteron of control has the form ⎛T ⎞2 æ[ u ] = ⎜ ∫ u (t )dt ⎟ ⎜t ⎟ ⎝0 ⎠ (4.4) Avoiding vigorous mathematical expressions, assume that b(1) = b(2) = b(3) = , a2(2) = , a1(3) = −2 , a2(3) = The normed fundamental matrices of solution of the homogeneous parts of systems (4.1)-(4.3) have the following form respectively ⎛ et −t1 − 1⎞ ⎛ t − t0 ⎞ X [ t , t ] = , X 1[t , t0 ] = ⎜ ⎜ ⎟, ⎟ 1 ⎠ et − t1 ⎠ ⎝0 ⎝0 ⎛ 2et −t2 − e 2(t − t2 ) X [t , t2 ] = ⎜ t −t 2( t − t2 ) ⎝ 2e − 2e e 2(t − t2 ) − et −t2 ⎞ ⎟ 2e 2(t −t2 ) − et −t2 ⎠ For the matrix H k [tk , t ] = X k [tk , t ]B (tk ) (k = 1, 2,3) , the following is obtained t −t ⎛ h11 (t1 , t ) ⎞ ⎛ t −t ⎞ ⎛ h21 (t2 , t ) ⎞ ⎛ e − 1⎞ H1[t1 , t ] = ⎜ ⎟=⎜ ⎟ = ⎜⎜ t − t ⎟⎟ ⎟ , H [t2 , t ] = ⎜ ⎝ h12 (t1 , t ) ⎠ ⎝1 ⎠ ⎝ h22 (t2 , t ) ⎠ ⎝ e ⎠ 2( t − t ) t −t ⎛ h31 (t3 , t ) ⎞ ⎛ e − e ⎞ H [t3 , t ] = ⎜ = ⎟ ⎜⎜ 2(t −t ) t −t ⎟⎟ ⎝ h32 (t3 , t ) ⎠ ⎝ 2e − e ⎠ According to (3.5), introduce the following notations ⎛ V (30) V (30) ⎞ ⎛ e(2 − e) e ⎞ V (t3 , t0 ) = ⎜ 11(30) 12(30) ⎟ = ⎜ ⎟, V22 ⎠ ⎝ 2e(1 − e) e ⎠ ⎝ V21 V R Barseghyan / Control of Stage by Stage 38 ⎛ V (31) V (t3 , t1 ) = ⎜ 11(31) ⎝ V21 ⎛ V (32) V (t3 , t2 ) = ⎜ 11(32) ⎝ V21 V12(31) ⎞ ⎛ e(2 − e) 2e(e − 1) ⎞ ⎟=⎜ ⎟ V22(31) ⎠ ⎝ 2e(e − 1) e(3e − 2) ⎠ V12(32) ⎞ ⎛ e(e − 2) e(e − 1) ⎞ ⎟=⎜ ⎟ V22(32) ⎠ ⎝ 2e(1 − e) e(2e − 1) ⎠ Now following (3.7), denote ⎧t1 − t , for t ∈ [t0 , t1 ) h11 (t1 , t ) = ⎨ ⎩0, for t ∈ [t1 , t3 ] ⎧1, for t ∈ [t0 , t1 ) h12 (t1 , t ) = ⎨ ⎩0, for t ∈ [t1 , t3 ] ⎧0, for t ∈ [t0 , t1 ) ⎪ h21 (t2 , t ) = ⎨et2 −t − 1, for t ∈ [t1 , t2 ) ⎪0, for t ∈ [t , t ] ⎩ ⎧0, for t ∈ [t0 , t1 ) ⎪ h22 (t2 , t ) = ⎨et2 −t − 1, for t ∈ [t1 , t2 ) ⎪0, for t ∈ [t , t ] ⎩ ⎧⎪0, for t ∈ [t0 , t2 ) h31 (t3 , t ) = ⎨ 2(t −t ) t −t ⎪⎩e − e , for t ∈ [t2 , t3 ] ⎪⎧0, for t ∈ [t0 , t2 ) h31 (t3 , t ) = ⎨ 2(t − t ) t −t ⎪⎩2e − e , for t ∈ [t2 , t3 ] According to (3.8), we have integral relations T T t0 t0 ∫ h1 (t ) u (t ) dt = c1 , ∫ h2 (t ) u (t ) dt = c2 , (4.5) where h1 (t ) = V11(31) h11 (t1 , t ) + V12(31) h12 (t1 , t ) + V11(32) h21 (t2 , t ) + V12(32) h22 (t2 , t ) + h31 (t3 , t ) , h2 (t ) = V21(31) h11 (t1 , t ) + V22(31) h12 (t1 , t ) + V21(32) h21 (t2 , t ) + V22(32) h22 (t2 , t ) + h32 (t3 , t ) , c1 = x1 (t3 ) − V11(30) x1 (t0 ) − V12(30) x2 (t0 ) , c2 = x2 (t3 ) − V21(30) x1 (t0 ) − V22(30) x2 (t0 ) Assume that the following numerical values are given t0 = 0, t1 = 1, t2 = 2, t3 = , and V R Barseghyan / Control of Stage by Stage 39 ⎛0⎞ ⎛3⎞ x(0) = ⎜ ⎟ , x(3) = ⎜ ⎟ ⎝ ⎠ ⎝ 2⎠ Solving problems (4.4) and (4.5) as the problems of moments, the optimal control action are obtained ⎧0,997012 − 0,841126t , for t ∈ [t0 , t1 ) ⎪ u (t ) = ⎨−0,841126 + 2, 71016e− t , for t ∈ [t1 , t2 ) ⎪ −2 t −t ⎩−5,87287e − 2, 71016e , for t ∈ [t2 , t3 ] and the value of the criteron of quality is æ ⎡⎣u ⎤⎦ = 0,55481 Substituting u (t ) into equation (3.4), written for systems (4.1)-(4.3), the optimal motion of object for every interval of time will be obtained REFERENCES [1] [2] [3] [4] [5] [6] [7] Arutyunov, A.V., and Okoulevitch, A.I., “Necessary optimality conditions for optimal control problems with intermediate constraints”, Journ of Dynamical and Control Systems, (1) (1998) 49 -58 Matveikin, V.G., and Muromcev, D.U., Theory of Energy-Efficient Control of the Dynamic Conditions of Production-Technological Purpose Apparatus, Moscow, 2007 (in Russian) Pontrjagin, L.S., and others, Mathematical Theory of Optimal Processes, Nauka, Moscow, 1969 (in Russian) Krasovski, N.N., Theory of Movement Control, Nauka, Moscow, 1968 (in Russian) Zubov, V.I., Lectures on the Theory of Control, Nauka, Moscow, 1975 (in Russian) Vereschagin, E.F., “Investigation methods of the flight regimes of variable mass devices”, Mechanics of the programmed movement of device, (2) Perm, 1972 (in Russian) Barseghyan, V.R., “About an optimal control problem of changing linear systems in discrete steps with phase restrictions during the intermediate moments of time”, Scientific notes Yerevan State University, (2002) 118-119 (in Russian) ... / Control of Stage by Stage 32 systems and stage by stage changing linear dynamic systems are stated and investigated in [2] and [7] The control and the optimal control problems of linear dynamic. .. dynamic systems changing stage by stage are stated and investigated in this paper PROBLEMS' DEFINITIONS Consider a control process, dynamics of which is described by the phased changing linear. .. operation of transposition Then equation (3.8) can be written V R Barseghyan / Control of Stage by Stage 35 T ∫ h′(T , t )u (t )dt = C i i (i = 1,K , n) t0 So, the condition of controllability of system