Çok girişli sistemlerde sınırlandırmalı sonlu zaman kontrolu

Abstract

ÖZET Sonlu zaman kontrol problemi, belirli bir başlangıç anında, belirli bir başlangıç durumunda bulunan bir sistemi, istenilen bir varış durumuna en küçük adım sayısında iletecek giriş değer dizisinin belirlenmesi problemi olarak tanımlanır. Giriş değer dizisinin belirlenebilmesi sistemin kontroledi lebi I ir olmasını gerektirir. Giriş ve durum değişkenleri sınırlandırılmamış tek girişli ve kontroledi lebi I ir sistemlerde sonlu zaman kontrol probleminin çözümü tekdir. Ancak çok girişli sistemlerde, problemin çok sayıda çözümü ile karşılaşılabilir. Giriş ve durum değişken lerinin sınırlandırılması halinde ise, sistemi varış durumuna ulaştırmak için gerekli adım sayısı artabileceğinden, tek girişli sistemlerde bile, sonsuz sayıda çözüm bulunabilir. Tezde ilk olarak, sonlu zaman kontrolü, giriş ve durum sınırlandırmaları göz önünde bulundurulmadan tanıtılmış ve çok girişli sistemler için problemin çözümüne ilişkin koşullar belirlenerek, sonlu zaman kontrollü sistemlerin zamanda. ayr ık lineer regülatör özellikleri tartışılmıştır. İkinci aşamada, giriş ve durum sınır I andırma I ar inin bulunması halinde, problemin geometrik yorumu yapılmış ve bu sınırlandırmalar sırasıyla değerlendirilerek, çok girişli sistemler için. sıfırdan farklı varış durumlarına da erişilmesini sağlayan bir sonlu zaman kontrol algoritması türe tilmiştir. Ayrıca, sınırlandırılmış girişler için sonlu zaman kontrolü i le minimum zaman kontrolü arasındaki ilişkiye değinilerek, iki probleme ait çözümlerin özdeşleştiği durumlar tartışılmıştır. Türetilen kontrol algoritması, çalışmanın son bölümünde, gerçeğe uygun bir model kurabilmek ve ölçüm yapabilmek amacıyla, analog ve sayısal bilgisayarlar hibrit çalıştırılarak, çeşitli sistemlerin sonlu zaman kontrol l arı na başarıyla uygulanmış ve elde edilen sonuçlar irdelenmiştir. vı ı

SUMMARY CONSTRAINED DEADBEAT CONTROL FOR MULTI-INPUT SYSTEMS Deadbeat control can be defined as the determination of a control sequence, so that the control process can be taken from the initial state xo to a desired state xt in a minimum number of time steps. As it is obvious from this description, deadbeat control problem can also be represented as the discrete time optimal control problem. Early contributions on the dead-beat concept, depending on the input-output relations of the systems were given by Ragazzini & Franklin (1958). When the input-output relations were of concern, the problem is defined as the determination of a discrete controller that would force the system output to reach to the system input in a finite time and follow it without any steady-state error. The state space approach to the topic was given by Kalman (1960) who solved the problem of transfering the state of a single input samp led- da t a system from its initial state to the origin in a minimum number of time steps using a linear state feedback. So, the problem was treated as the discrete time optimal control problem. In solving this problem, Kalman has also introduced the concepts of controllability and observability. Then, many authors contibuted to the subject and various studies such as deadbeat control of mul t ivar iable systems, input and state constrained deadbeat control, inaccesible state deadbeat control, deadbeat control of time-varying systems deadbeat control in non-linear systems are still under research. This study deals with input and state constrained deadbeat control problem and is divided into 4 chapters as the following: Chapter 1 : Introduction Chapter 2 : Representation of The Deadbeat Control Problem and Its Properties Chapter 3 : Input and State Constrained Deadbeat Control Problem Chapter 4 : Modelling of The Constrained Deadbeat Controlled on a Hybrid Computer In the first chapter, studies concerning the problem are sumnar i zed and some fundamental notions about the problem are given. Chapter 2 is devoted to the representation of the deadbeat concept and its properties. In this chapter, unconstrained deadbeat control problem is investigated and its fundementals are given in Section 2.1. Based on the controllability criterion, a controllable system can be taken from a given initial state to a desired final vi i istate in u time steps, where u. is the contol labi I i ty index of the system. When the input and state variables are unconstrained, u defines the least number of time steps, required for the determination of the input sequence that will force the system to a desjred final state. If a system that is taken from an initial state xo^x^ko) to a final state xt=x>(kt) in u time steps is considered, it is obvious that, the minimum number of the time steps required to reach from any state x(k) (ko<k<k«) to x« is less than u. Thus, u can be declared as the greatest of the least number of the time steps required for this system to reach from various initial conditions to a definite final state. As long as the input and the state variables are unconstrained, the deadbeat control can be defined as the determination of the input sequence that will force the system from xo to xi in at most u time steps, and the input sequence Up should satisfy the equation, -> where [ZM] is the control labi I i ty matr ix and the vector XM is composed of the initial and the final states. For single input systems, the contol labi I i ty index is equal to the order of the system n, and the deadbeat control can be realized in n time steps. As it is mentioned in Section 2.2, when multi-input systems are of concern, the problem is some more complicated, since the solution deals with the selection of n linearly independent columns of the contol labi I i ty matrix. There are many selection procedures and the deadbeat solutions of the mul t i-input systems are not uniqe. The linearly independent columns of the controllability matrix can be selected by defining various transformation matrices. The order of the transformat ion matr ix and values of its elements define the selection procedure. It is usually desired for the system to stay at the final position after it has reached to it. For this purpose, the final state has to be an equilibrium state of the system to be control led. Transfer ing the states of a system from any initial state to the origin in a minimum number of time steps using a linear state feedback can be formulated as a discrete linear regulator problem. Deadbeat control in mul t i-input systems as a linear state feedback is investigated in Section 2.3. Since the solution to the multi-input case is not unique, various forms of state feedback controllers can be found in literature. It is necessary to use an ordered selection of the linearly independent columns of the control labi I i ty matr ix in order to obtain a state feedback gain matrix and the control law is given by u(k) - [F] x(k) where [F] is the feedback gain matrix. The closed loop system matrix [Qk] thus obtained has the following properties: 1- [Qui is a ni Ipotent matrix of which the index of nilpotency is the system ixnilpotency index u, 2- The eigen values of [Qk] are zero, 3- The matrix [Qk] is similar to the Jordan matrix, composed of r nilpotent Jordan bloks where r is the number of inputs. In practice, because of the physical constraints, the elements of the input sequence and the the states of systems can not take values above some saturation limits. For this reason, the input sequences for the deadbeat control has to be calculated taking these limits into consideration. Chapter 3 is devoted to the input and state constrained deadbeat control problem. When the input and state variables are constrained, the desired forcing effort applied to the process is restricted by the saturation limits and the minimum number of time steps required to take the process from xo to xt would be more than u which is valid for the unconstrained case. Input and state constrained deadbeat control problem can be defined as the determination of a control sequence, so that the control process can be taken from the initial state xo to a desired final state xt in a minimum number of time steps and, for all the elements of the input sequence and the system states under the following constraining conditions Umin<Uj(k)<Umax ( j-1,2,.., r ) and [D8] x(k) < x8 where x*s is the state constraint vector and [D8] is the state constraint coefficient matrix. If q is represented as the minimum number of the time steps to take the process to its final state, under the above conditions, the inequality u<q wi 1 1 always be valid. So, the input sequence Uq has to satisfy the equation -[Zq] UQ where [Zq] is obtained by extending the controllability matrix [ZM] in columns and Xq is composed of the initial and the final states. In Section 3.1 some fundamental geometrical aspects of the input and state constrained deadbeat control problem are given as the following: The input constraint describes a (q.r) dimensional hypercube in the Uq input space. The input and the state constraints together represent a (q.r) dimensional hyperprizm (the shape of which is identified by the hyper sur faces described by the state constraints). If all the input signals which are solved for u time steps lie in this hyperprizm, the saturation limits can be ignored. If the input signals exceed the saturation limits, then the corresponding input step needs to be broken into smaller steps, so that each step amplitude is equal to or smaller than the saturationlimit. Thus, it takes more steps for the saturated system to reach to the final state. Since, a large input step can be broken into smaller steps in an infinite number of ways, there exists an infinity of optimal solutions to this control problem. In the presence of saturation, when it is assumed to reach the final state in q time steps, where u<q, the solutions of the problem describe a hyperplane in the input space. If this hyperplane, which may be refered as the solution plane, does not intersect the hyperprizm formed by the saturation constraints, the above assumption is not valid and the process will not be able to reach the final state in q time steps, but the process may be made to reach the final state in q+1 or more steps. If the solution plane intersects the hyperprizm, there is one or an infinite number of solutions to this optimum-design problem. Input and state constrained deadbeat control problem can be transformed and solved by the Linear Programming or Simplex method. Another solution method for the input constrained deadbeat control problem can be obtained by exploiting the geometric approach breifly described above. Depending on this second method, an input and state constrained deadbeat control algorithm is derived in Section 3.2. In the algorithm, multi-input systems are taken into consideration with zero or non-zero final states. In the presence of input saturation, the algorithm depends on the principal that: If an input signal exceeds a prescribed input saturation limit, the value of this saturation limit can be given to this input variable under the consideration that the input variables following this variable in time lie in the hypercube described by the input saturation constraints. Although the number of time steps required to solve the problem is unknown at the beginning, there is a possibility to arrive at the solution in q=u time steps such that all the elements of UM may satisfy the input conditions. _Therefore, the set of simultaneous equations given by Xq=-[Zq] Uq is transformed into its upper triangular form by using the first n linearly independent columns of the matrix [Zq]. Then, the elements of the input vector are calculated begin i ng from the n.th equation. As long as the input variables thus calculated do not exceed an input constraint, the calculation continues recursively until all the elements of the input vector are obtained. But, when any one of the elements of the input vector exceeds a saturation limit, the a Igor i t hm assigns this saturation value to this input variable, since the following input variables which are calculated before this variable lie in or on the hypercube described by the input constraints. Then, the preceeding input variables are recalculated by extending the number of time steps if it is necessary. For this purpose, a new set of simultaneous equations is formed beginning wi th the equation m which has been used to calculate the input vector element exceeding the saturation limit. Then, the input variables are calculated from the new set of equations using the same method. When the recalculated input variables and the input variable which has been assigned the saturation limit value are applied to the m-1.th equation, a preceeding input variable can be calculated. These calculations continue until all the elements of the input sequence UQ are obtained. It is seen that the algorithm is composed of several loops in which the same calculations are being performed. When multi-input systems are taken into consideration, it is necessary to describe XIthe linearly independent columns of the matrix [ZqJ in every iteration of the algorithm. The coefficient of x» in Xq depends on the number of time steps q which may take^dif ferent values in every^ iteration. Therefore, if the final state xi is non-zero, then Xq should be evaluated wi th the suitable q value. The geometrical aspects of the algorithm and the detailed mathematical formulation are given in sections 3.2.1.1 and 3.2.1.2 respectively. The flowchart of the algorithm and an example take place in Section 3.2.1.3. In Section 3.2.2, the input constrained deadbeat control algorithm given in Section 3.2.1 is extended to the case where the state variables are also constrained. When the state constrained case is investigated it is seen that the state conditions rely on the summations of the input variables. So, it is obvious that, there may be cases where the state conditions are satisfied by only constraining the input variables. Therefore, the problem can first be solved in the absence of state constraints. If the resultant trajectory is not violating the state constraints, it means that the problem has reached to a solution. But if one of the state conditions is not satisfied, then the limit value is given to this constraint and an new equation is obtained. The previous set of simultaneous equations is extended by adding this equation as the n+1.th equation and the input constrained dead-beat control problem is solved again by using the algorithm given in Section 3.2.1. If there are st i 1 1 unsatisfied state constraints, equations dealing with these constraints are added to the new set of equations formed as described above. When more than one state constraints are unsatisfied, there is a posibility that, just by assigning their limit values to some of them, the others would not violate their constraints. Therefore, in the presence of more than one violated state contraints, it is necessary to extend the equation set by using different combinations of state constraint equations (begining with the one that consists of the minimum number of equations). In the last section of Chapter 3, the relations between input constrained deadbeat control and continuous minimum time control problems are investigated. Continuous minimum time problem can be described as the determination of an input variable that would take the continuous system from the initial state xo^x^to) to the final state Xf-x'ttt) in a minimum time tmin while the following cond i t i on Umin < Ui(t) < Umax (j=1,2 r) is satisfied. It is seen that continuous minimum time problem and discrete time input constrained deadbeat control problem coincide with each other depending on the sampling period T. Let us consider that input constrained deadbeat control problem is solved for various values of sampling period T, where the period T is changed with small perturbations. Then it is seen that the same number of time steps is required for some succesive group of T period values. When the minimum T period of one of these groups is considered, it will be seen that the arrival time will be equal to or very near to tmi n ? Xİ İIn Chapter 4, the algorithm given in chapter 3 is simulated on an analog and a digital computer where the system to be controlled is set up on the analog computer and the controller part (or the algorithm) is run on a digital computer. First, a second order system with single input, then a second order system with two inputs, and at last, a third order system with two inputs are modelled on the analog computer. The instrumentation of the hybrid set up is given in Section 4.1.1. The discrete parameters of the systems modelled on the analog computer are estimated by the `least square estimation` method as it is explained in Section 4.1.2. Systems are modelled on the analog computer by the ordered connections of the integrator, amplifier and potant iometer sections of the computer. The control algorithm is program-tied on the digital computer by the `basic` programming language, and the `quick basic` version is being used. The continuous system parameters, the sampling period, the input and state constraints, the initial and the final states are given to the program as the inputs. In order to run the estimation algorithm, n+r measurements are made. The inputs used for the first n+r measurements are calculated theorat ical ly for the deadbeat control of the system. One requirement of these input signals is that they should ex ite all the modes of the process sufficiently. Therefore, the constraints are chosen smaller than the process would accept. The states of the system measured at the n+r.th sampling period (where system parameters are estimated) are given to the control algorithm as the initial states, and the input that should be applied to the estimated system at the following sampling instant is calculated. In order to avoid from the distortion effects, the control algorithm is run in every sampling period. XIII