Control Theory in Linear Algebra - Coursework Example

Control Theory in Linear Algebra Student’s Name: Institution: Control Theory in Linear Algebra Introduction Multidimensional system analysis (n-D) has received extensive importance over the past few decades. However, the modeling of a system of this kind is still among the underlying development to be completed. Generalizing 1-D state space concept is one approach that has to be generalized. These generalizations, as well as their corresponding results, have been limited to some extent due to some major differences between the algebraic structure of the n-D system and 1-D system. The concept of the past, present and future is inherent to the classical approaches in the operating time domain. There is no natural dialogue to this in the case of n-D. Therefore, this means that there are possibilities that are sometimes infinite existing for state updating equations. Any imposed parts will also be having a number of present states at any given time. The present will be indicated using a line in the 2-D case as well as the n-D case, a dimension hyperplane n-1. Therefore, it is possible to distinguish two notions of state. It is important to note that each of the two hyperplanes of dimension n-1 results to a global state which is an analog to the state variable of 1-D. On the other hand, every point in the hyperplane makes up a local state. Therefore, this is an indicator that the two kinds of system properties, global and local can be characterized. The figure below shows the behavioral modeling of a dynamic system. Figure 1: Behavioral Modeling of a Dynamic System There are many results in the theory of 1-D system over a field K which are related to Euclidean character K[z]. In the case that actually belongs to the study of communicative Noetherian rings, this case, such tools not available anymore. Among other differences, these differences clearly indicate that other approaches are also desirable. An n-D system is an approach that was introduced by Willems (1983) in his 1-D study cases and was later generalized to 2-D by Rocha (1990). A dynamic system can be seen as an entity interacting with its immediate surroundings via a variable known as external variables. There are system laws in place to govern the relationships which exists between the external variables of the system and also leads to the rise of a family admissible trajectories for the external variables. System behavior is the term used to refer to the admissible system trajectories set. A representation of this system is constituted by a mathematical description of the system. It is easier and convenient in many instances to work with representations containing external and auxiliary variables. The extra variables are known as internal variables. They may also be called latent variables at times (Willems, 1991). The state variables give the typical examples of internal variables that are always introduced for drafting the dynamical equations as first degrees equation of a given system. The following are the differences between the classical models and the behavioral models: There are instances or situations that it is not possible or convenient for having a prior knowledge of the external system variable division into outputs and inputs. The behavioral approach can handle these situations. This is also applicable in the classical context whereby such a priori is assumed of given. There is a set (system behavior) influencing the dynamical system. The dynamic system is characterized by this set instead of a function like an input-output map. The following example can be used for illustrating the above idea: It is always of interest to measure the signal velocities in directions of x, y, and z in seismology. This is important because formations of different characteristics are penetrated by seismic waves, with different velocities. Let a 3-D velocity filter be in such a manner that it passes the signal whose velocities are within a certain region in the (f.k) space that is cone shaped. Whereby f represents the frequency and k represents the 2-D wave number consisting of two components kx and ky. This is to be given by: The geographical structure determines the system’s behavior. Which an arrange of velocities v = (vx. vy, vz) that are compatible with geography law and can also be given by: B ={(vx, vy, vz) : IR3 → IR3│v falls within the cone-shaped region given} The Roesser Model This model enables the generality of specific analysis structure and design, and one-dimensional state design result space systems by enabling its model’s local state to be divided into a vertical and horizontal state that are propagated respectively, vertically and horizontally by difference equations of the first order[Gop10]. We can find the extension of this model to n-D in Kaczorek (1992) study. With the interest that is growing in the singular system, n-D (2-D) generalization of Roesser models can be done as follows (Kurek, 1989),[Lew92]: Let’s consider hyperbolic partial differential equation With the following initial value of conditions of boundary: T(x, 0) = f1(x) T(0, t) = f2(t) Let If xh (i.j) = T (i – 1.j) and xv (i. j) = T(i, j) is defined, then by rewriting the form Where We will then obtain the Roesser model: Controlled Invariant According to Basile and Marro (1992), the state space controlled invariant presentation for certain systems is considered a subspace in a way that it is possible to control the system in such a manner that all times the state is inside the subspace if the system’s state is initially available in the subspace. In defining controlled invariant subspace, we will consider a linear system which following differential equation clearly indicates. X(t) = Ax(t) + Bu(t). In which the system’s state and u(t) ∈ Rp is the input assigned by x(t) ∈ Rn The sizes of matrices A and B are n × n and n × p in the same order. If for any x(0) ∈ V, there is an input u(t) in a way that x(t) ∈ V for all non-negative t, then A subspace V ⊂ Rn is considered to be a controlled invariant subspace. According to Gosh, 1985, if and only if AV ⊂ V + Im B will a subspace V ⊂ Rn be considered a controlled invariant (Ntogramatzidis, 2012). History of Dimensional Spaces In mathematics and physics, bi-dimensional space or two-dimensional space refers to the physical universe's planner projection geometric model. The lengths and widths are the two dimensions, and both directions are lying on the same plane. The 'n' real number sequence is understood as a location in dimensional space. The set of all locations of such kind is referred to as bi-dimensional space or two-dimensional space when n=2. It is usually thought as Euclidean space. The books of Euclid's elements from I through to IV and VI were majorly concerned with two-dimensional geometry and developed notions as the similarity of shapes, equity of areas and angles, Pythagorean theorem, parallelism, the three cases making the triangle equal, the sum of angles in a triangle and several other topics. The planes were later described by a coordinate system uniquely specifying each point in a plane by a pair of numerical coordinates that are signed spaces from the point of perpendicular directed lines that are fixed. Measured in the similar unit of length. Each reference line is referred to as an axis or a coordinate axis. On the other hand, the point where they meet is referred to as origin. It is in 1637 that the idea of this system was introduced independently by Pierre de Fermat and in writing by Descartes. Fermat later started working on two-dimensions but never published the discoveries. In their treatment, the two authors both used single angle axis and had a variable length measurement as references to the axis. Later on in 1649 after Descartes was translated into Latin, the concept of using a pair of axes was introduced. Several concepts were later introduced by these commentators as they were trying to make clear the different concepts that were contained in the work of Descartes. The plane was later thought as a field where two points except zero (0) could be multiplied and divided. It was referred to as a complex plane. Because of its use of Argand diagrams, the complex plane may sometimes be referred to as Argand plane. Even though they were first described by Mathematician and land surveyor Casper Wessel (1745-1822) who was also a Norwegian-Danish, they were named after Jean-Robert Argand. The diagrams developed by Argand are today frequently used for plotting the positions of zeroes and the poles as a function in the complex plane. Geometry In analytic geometry, mathematics, which is also referred to as Cartesian geometry, describes all points in two-dimensional space through the means of coordinates. There are two perpendicular axes crossing one another at the origin. They are always labeled x and y. Relative to these axes, the positions of points in two-dimensional spaces is given through an ordered pair of real numbers. All the numbers give the distance of points from the origin as measured along the axis that has been given, which is also equivalent to the axis of that point from the other axis. The polar coordinate system is another widely used coordinate system that specifies a point in terms of distance from its angle and origin relative to a right reference ray. Linear Algebra Linear Algebra consists of another mathematical method of viewing two-dimensional space, where the thought of independence is critical. The plane consists of two dimensions because the lengths of a rectangle depend on upon its width. The plane is described to be two-dimensional using the technical language of linear algebra. This is because all points in the plane are described by a linear combination of two vectors which are independent. The Dot product of 2 vectors A = [A1, A2] and B = [B1, B2] is elaborated by the equation that follows: A.B = A1B1 + A2B2. This vector is viewed to be narrow with its directions being in the arrow point directions and magnitude being in its length. ║A║ denotes the magnitude of vector A. At this point, the Euclidean A and B vectors dot product is delimitated by A.B = ║A║ ║B║ cos θ in which the angle between A and B is θ. A.A = ║A║2 is the dot product of vector A by itself. This results to ║A║= √A.A, which is the formula for Euclidean length for the vector. Algebraic System Theory In the past 15 years, algebraic system's theory has been greatly advanced. This has been because of the behavioral way to deal with frameworks and control theory that was initially presented in the 1980s by J.C Willems (Zerz, 2006), and which was also turned out to be especially advantageous for methodologies of algebra based on studies sets of solutions instead of representing equations. It is also important to note that it is not dividing the system variables into differently treated classes. However, in the 1960s, V. Palamodov[Pal70], and B. Malgrange[BMa64] and others started studying equations of the system of linear partial differentials with the use of algebraic tools like homological methods and module theory. They came up with what is today known as the algebraic analysis approach. A similar paper by Oberst (1990) established a link between the two approaches, resulting in a meaningful understanding of both. The lively research activity in the area of the multidimensional system was stimulated by this. It also led to behavioral and algebraic system theory in general. Abstract Linear Systems Let A be a left D-module and D a ring consisting of 1 which is necessarily not commutative. An example will be: for matrix R ∈ D g×q given, the abstract is considered to be a linear system. B = {W ∈ Aq | Rw = 0}. B being a letter has been selected to be the behavioral approach in allusion. Letter A and D should be thought as a set of signals and a ring of differential operators respectively. The ring of differential follows up on them. This is to say that A conveys a D-module structure amount to the requirements in a manner that someone is able to apply any operator d∈D to any signal and a ∈ A for obtaining fresh signal da ∈ A. In a similar manner, the formulation of Rw from above will be an element of Ag that is well defined. According to Malgrange (1964) important observation, B is considered to be Abelian group that is isomorphic to the group of D- homomorphisms (with respect to addition) from M : = D 1xq / D 1xg R to A. This is to say, B ≅ HomD (M, A). Proving the results is not difficult, but its significance depends in the fact that it attracts focus on M as an algebraic object. This object is referred to as the contravariant function HomD (.,A) and the system module that turns into Abelian the left D-modules. Someone may say that an injective cogenerator is D-module A [Lam99] when factor F :=HomD (.,A) reflects and preserves exactness, which is, a sequence of D-homomorphisms and D-modules. Multidimensional System Let the partial differential operators ring be denoted by D=K[∂1,...,∂n] with constant coefficients (whether complex or real), and let A= C∞(Rn, K). A is known here to be injective cogenerate, and D is communicative[Obe90]. A linear system refers to the smooth set of homogeneous arrangement of linear consistent coefficient PDE that is otherwise called a multidimensional framework or linear, move invariant framework. The two properties that takes after are basic in multi-dimensional framework hypothesis: - If there were no free variable inputs, If there were no free variable inputs, B becomes autonomous. This similarly or equivalently happens when 0 ≠W ∈ B does not exist with compact support[Pil99] If B will not be controllable if it is parametrizable. That is it will be having a picture or image representation represented by B = {Mℓ | ℓ ∈ Al}and for some M ∈ D q × l similarly for all sets which are open U1, U2 ⊂ Rn and w1,w2 ∈ B with U1 ∩ U2 = ∅. W ∈ B[Pil99] W(x) = w1(x) if X ∈ U1 w2 (x) if X ∈ U2. The characterization of controllability and autonomy are available regarding the framework module. M = D1×q/D1×g R (Pommaret & Quadrat, 1998; Pommaret & Quadrat, 2000): If and only if M is torsion is when B is considered to be autonomous. This is to mean any presentation matrix R of B is having a column rank which is full. If and only if M is torsion-free will B be controllable. This is to say, any presentation matrix R of B considered to be a left syzygy matrix, which is, the left kernel is generated by the rows of R, {z ∈ D1×q | z M = 0}of some M ∈ Dq ×l One-Dimensional (1D) Time-varying System Let the ring of linear ordinary differential operators be denoted by D = K [d/dt] with coefficient in the field K of meromorphic functions. According to Cohn (1971), it is not a commutative ring, but it is an ideal domain for simple left and right principal. There is an unimodular matrix, U, V for every R ∈ D g × q in such a way that URV is diagonal. It is considered to be a non-commutative analog that is due to Jacobson. It is an analog in Smith form[Coh71]. One must work with A = C∞ae(R, K) to cope with the coefficient singularities, which are considered to be the set of all smooth functions apart from a finite (also referred to as the discreet) several numbers of points. Different approaches can be attained from the following authors and books (Fröhler & Oberst, 1998; Ilchmann & Mehrmann, 2005; Pommaret & Quadrat, 1999). A is again referred to as an injection cogenerator as described by Zerz (2006; 2005). It will gain have characterization of autonomy and controllability illustrated below, which are considered to be generalization of characterization known pretty well in the time-invariant (constant-coefficient) case: If only B is capable of being represented by full ranks squire matric, then it will be autonomous, that is, without inputs. The notion of ranks brings meaning in the usual way since D can be firmly enclosed into a skew field of fractions. If it can be presented by a representation matrix that is right invertible does B becomes controllable. This means that it has an image representation. In fact, any representation of full-row-rank of a controllable system must be inevitable. One-Dimensional Parameter Dependent system Let D = R ∈ D g×q and K[p1,...,pN][d/dt] Where p= (p1,...,pN) and is the system parameter vector. The family of ODE systems will be described by R: we will obtain R|p=p0 ∈ K[d/dt]g×q for the every choice of p0 ∈ K N, therefore a one-dimensional system (n – 1). B|P = P 0 = {W ∈ Aq|R|p = p 0 W = 0} Assume that R initially had a column rank which is full. Then the family system B will be referred to as genetically autonomous. The rank drop in R may be caused by specific parameter constellation. This parameter values Po is determined by this whereby the system of framework B|p = p0 will lose autonomy. It is worth noting that even though the representation matrix ranks will be the similar for all values of the parameter, controllability may be destroyed by special parameter constellation. We will assume for this that R|p = p0 has a row full raw rank for every P0 ∈ KN. If and only if R | p = p0 is right invertible over K[d/dt] will B|p=p0 be able to be controlled. If R is invertible over K(p1,...,pN)[d/dt], then the system B will be said to be generically controllable. This is an implication that for nearly all P0 ∈ K N, B|p = p0 will be controllable. B|p = p0 will be precisely controllable for every Po not within the algebraic variety. V – V (ann (N) ∩ K[p1,...,pN]), in which N : - Dg / R D q, thus ann (N) – {d ∈ D | ∃X ∈ Dq× g : RX = dI}. One would, however, like to avoid the annihilator deal computation when considering the utilization of wider examples. This is a heuristic technique used to detect the consists of important constellations of parameter in examining generic controllability over K[p1,...,pN][d/dt] and to keep track of every denominator which appears in the computation. The outcome may be conservative in a manner yielding more contender for controllability that wrecks parameter constellations than required or important. The same is, be that as it may, valid for the strategy using annihilator perfect, it is the best possible subset of V that is always the arrange of points whereby controllability is lost by the system. Similarly, the heuristic techniques can likewise apply to rationally (instead of polynomially) parameter-dependent/subordinate families of systems. Implementation Controllability and autonomy tests of the multidimensional system are always implemented in a singular[Pfi05], library control, (Zerz & Levandovskyy, 2005), is accessible with SINGULAR from form/version 3.0 onwards. There is a better order provided regarding controllability and autonomy degrees (Pommaret & Quadrat, 2000; Pommaret & Quadrat, 1998) and also more output like flat outputs, parametrization, etc. a methodology understanding the heuristic strategy for constellations of important parameters depicted previously is also included in the SINGULAR control library. The major aim of computation is user friendliness and efficiency, specifically, in requiring minimal pre-knowledge of algebra, its target audience comprises majorly of control theorists. MAPLE package OREMUDULES offer a broader functionality[Chy07], which concentrates on calculations that are non-commutative, and that were the very first algorithm implementation for computational arrangement of issues identified with control in the methodology of algebraic examination. It has been projected that in the commutative case, in which there is the possibility of direct control, control.lib will outdo OREMODULES with examples that are large because of the high SINGULAR efficiency with basic computation, which is of the polynomial standard. Non-commutative computer algebra system PLURAL will be used for the extension of control.lib to variable coefficients that is coming forth[Lev03]. Linear Algebra As Applied in Control Systems Control System refers to devices which regulate, manage or command the behavior of other device or system. It can be thought of as having the following four functions: Measure, Compare, Compute, and Correct. The four functions of the control system are completed by the following five elements: Transducer, Transmitter, Controller, Detector, and Final Control Element[Roe75]. A control system can be practically implemented in embedded system with the use of a microcontroller or PLD’s. Control Theory, on the other hand, is an interdisciplinary mathematical and engineering branch which deals with the dynamic systems behavior. The major aim of control theory is to calculate solutions for proper correction action from the controller that leads to the stability of the system. The transfer function, which is also referred to as the network function, and the system function, is a mathematical tool used for defining the relationship between output and input. Classification of Control Theory 1. Sequential or Logical Controls: these are usually implemented using logic gates. It is commonly referred to as combination circuit. 2. Linear or Feedback Controls: the implementation of feedback or linear controls is always done using a combination of flip flops and circuit (Sequential circuit). 3. Fuzzy Control: these are attempts made for designing simplicity of logic with the utility of linear control. There are some systems or devices that are inherently not controllable. The figure below represents a negative feedback system[Tre12]. Analysis and Design of a Feedback System There are two approaches widely used for analyzing and designing feedback control systems: 1. State-space Approach: this approach is used for representing a multiple input multiple output system into a mathematical model[Tre12]. 2. Classical or Frequency-Domain Approach: this is an algebraic method commonly used to convert single input single output equation of simple differential into transfer function by transformation of system equation into frequency domain equivalent referred to as transfer function. Classical Approach The classical approach refers to the domain technique which incorporates mathematical tools like Laplace transforms. The following are steps taken in a classical approach. In time domain (system equation) is transformed into a transfer function of the frequency domain. What follows will be an undertaking computation, simplification, and analysis in the frequency domain. The results that have been obtained will be transformed back to time domain with the use of inverse transforms. Reference input Error Control Signal Disturbance Actual output desired Feedback Figure 2: Feedback System Illustration Advantages and Disadvantages of Classical Approach: the main advantage is that classical approach provides transient response information and rapid stability. The limited applicability is the primary disadvantage of the classical approach. Another disadvantage is that classical approach is only applicable to linear, invariant systems, which is capable of being approximated as such. State-Space Approach Under state space approach there is a presentation of system functions in the form of matrices. Here the presentation is not done in a single system equation. It is a method of modeling, designing and analyzing a wide range of systems in a unified manner and can be used in representing a nonlinear system. With the help of the state system, the modeling of the time-varying system becomes easy. 2-D (n-D) Linear Systems and LRPS For over three decades, the linear-space models for considered classes of 2-D (n-D) systems have been the subject of research because of the advantage of providing a simple, intuitive way for synthesis and analysis for the n-D system (Conte, Perdon, & Kaczorek, 1991). Even though the models of 2-D state-space may look similar with 1-D state space, between them, there is the existence of some differences. One of the main difference between 2-D (n-D) and models of 1-D state space is that in 2-D (n-D) cases, the models deals just with nearby state in spite of the worldwide state that jelly all past data as on account of 1-D. Along these lines, this implies some real ideas like controllability and dependability have to be defined for both global states and local state (Kaczorek, 1991). Based on the above descriptions, some basic properties such as performance bound, robustness, and stability are described. Furthermore, it will be clearly stated and shown that checking the n-D system properties is not always amenable to an algorithmic solution that is active because of the fact that they always have the characteristics of undecidable or NP-hard problems[Nto08]. Lastly, there are examples that are being used for illustrating properties and applications of presented models. State-Space Models of 2-D Linear System Considerable focus has been put on 2-D frameworks and systems in the past decade because of their practical importance (Kaczorek, 1985). The system has been intensively studied since the introduction of the linear state models in 1970s. Several state space 2-D models were introduced by Fornasini and Marchesini (Fornasini & Marchesini, 1978). Fornasini and Marchesini, Roesser, Kurek and Attasi are the models that have been extensively used to describe 2-D (n-D) systems and for investigation of various properties (Kurek, 1985). Givone-Roesser Model The state-space formalism was introduced by D. Givone R. Roesser introduced in 1972 for a linear interactive circuit that is a spatial system instead of a temporal system. The combination of individual cells is the interactive circuit. Each is identical in a pattern that is regular. This type of circuit that is widely utilized in the theory of logical circuit and automata[Kar92]. An interactive circuit is linear especially when outputs as well as inputs of each cell are in vector space forms over a common finite field, where a linear transformation is performed by each cell. It is important to note that a real field is utilized rather than a finite one in a case involving image processing. In a pragmatic point of view, the iterative circuit can be used for encoding, decoding and processing of images. u(h,k) x(h,k) xh(h,k) xh(h+1,k) xv(hk+1) y(h,k) Figure 3: Two-Dimensional Unilateral Iterative Circuit The figure above shows two-dimensional unilateral iterative circuit. The classification of the circuits may be done in two categories based on the directions of the signals that are flowing via every cell. These circuits can be bilateral and unilateral iterative circuits. The figure above depicts the 2D unilateral circuit. Fornasini-Marchesini Model G. Marchesini and E. Fonasini proposed an option formalism of exchange or digital elements of 2D, with the distinctive state-space acknowledgment. When the state-space representation was formulated by Givone-Roesser in view of the iterative circuit, Marchesini and Fornasini got statespace acknowledgment from the factorization of the n-D or 2D yield - input map taking into account the algebric perspective of Nerode comparability classes (Miri & Aplevich, 1998; Fornasini & Marchesini,1976). The Nerode equivalence classes can be demonstrated by: Letting u and v become the concatenation input-strings and be denoted by * than the inputs u and v will be similar in a manner of Nerode, denoted by u ~ v, when u and v are concatenated with the similar arbitrary-string w and the outcome of the outputs are similar despite the value of w. Let f be the output/input map. f : U →Y. Then for every u, v ∈ U, u ~ v if and only if f (u*w) = f (v*w) The minimum capacity of previous data required to completely determine the mapping from the future input to future output are considered to be the state at a particular period of time time (h,k) in the sense of Nerode, whereby the notation of the past and the future elaborated as Let T = Z x Z denote the grid (integer lattice) for 2D linear system[Bas92]. This should be characterized of equipped with the partial ordering of integer: (i; j) > (h; k) if and only if (i; j) ¸ (h; k) and (i; j) 6= (h; k); (i; j) = (h; k) if and only if i = h and j = k, (i; j); (h; k) If and only if i¸ h and j ¸ k. It is noted that for any t ∈ T, the time T is known as the past with regards to t if T isn’t more than or equal to t. The time T is therefore considered as the future with regards to t if T is greater than t. Thus, this Nerode state-space is always dimensional for the 2D system. This is what motivated E. Fornasini and G Marchesini to create a local state that is different and differentiable from the Nerode state, global state and it is the observation that gives the main difference between 2D and 1D systems[Lev90]. The local state only has information used only for computation of any state of interest at each recursions step, and thereby is able to attain system realization with the local state that is of finite-dimensional, but all past information will be provided by the Nerode state/global state every time (h,k). The global state is generally of finite dimension. (h,k) (h’,k’) For any two points (h’,k’) ≥ (h,k), a local state x ((h’,k’) dependent only not on the state x(h,k) but also on local states x(h+1,k),….,x(h’, k) and x(h,k+1),…,x(h , k’) as illustrated in the figure above. It will be important to enclose the global state space idea as follows since (h’k’) is considered to be the point of arbitrary on the two dimensional state space: XGlo, global state space refers to a 2D state space that comprises of all local state spaces on the vertical as well as horizontal axis. Particularly if h’ = h+1 and k’=k+1, then it will be possible to write the local state x (h’, k’) by updating the equations: x(h’, k’) = x (h + 1; k + 1) = A0x(h; k) + A1x(h; k + 1) + A2x(h + 1; k) + Bu(h; k) y(h; k) = Cx(h; k) Along with the boundary conditions and initial conditions x (0,0), X (0) ∆{x(0,j), x(i, 0) | 1≤ i ≤ M, 1 ≤ j ≤ N for some M,N ˃ 1}. From this equation, it is clear that the state x (h+1,k+1) is dependent on the state x(h,k) and therefore it is not considered to be the difference equation of the first order. Alternatively, the new state n can be defined as: n(h,k) = x (h,k +1) – A2x(h,k) then n (h + 1; k) = x(h + 1; k + 1) - A2x(h + 1; k) = A0x(h; k) + A1x(h; k + 1) + Bu(h; k) = A0x(h; k) + A1 [n (h; k) + A2x(h; k)] + Bu(h; k) = A1n (h; k) + [A0 + A1A2] x(h; k) + Bu(h; k) References Gop10: , (Gopinath, Kar, & Bhatt, 2010), Lew92: , (Lewis, 1992), Pal70: , (Palamodov, 1970), BMa64: , (Malgrane, 1964), Lam99: , (Lam, 1999), Obe90: , (Oberst, 1990), Pil99: , (Pillai & Shankar, 1999), Coh71: , (Cohn, 1971), Pfi05: , (Pfister, Greuel, & Schonemann, 2005), Chy07: , (Chyzak, Quadrat, & Robertz, 2007), Lev03: , (Levandovskyy & Schonemann, 2003), Roe75: , (Roesser, 1975), Tre12: , (Trentelman, Stoorvogel, & Hautus, 2012), Nto08: , (Ntogramatzidis, Cantoni, & Yang, 2008), Kar92: , (Karamancioglu & Lewis, 1992), Bas92: , (Basile & Marro, 1992), Lev90: , (Levy, Adams, & Willsky, 1990), Read More

The state variables give the typical examples of internal variables that are always introduced for drafting the dynamical equations as first degrees equation of a given system. The following are the differences between the classical models and the behavioral models: There are instances or situations that it is not possible or convenient for having a prior knowledge of the external system variable division into outputs and inputs. The behavioral approach can handle these situations. This is also applicable in the classical context whereby such a priori is assumed of given.

There is a set (system behavior) influencing the dynamical system. The dynamic system is characterized by this set instead of a function like an input-output map. The following example can be used for illustrating the above idea: It is always of interest to measure the signal velocities in directions of x, y, and z in seismology. This is important because formations of different characteristics are penetrated by seismic waves, with different velocities. Let a 3-D velocity filter be in such a manner that it passes the signal whose velocities are within a certain region in the (f.k) space that is cone shaped.

Whereby f represents the frequency and k represents the 2-D wave number consisting of two components kx and ky. This is to be given by: The geographical structure determines the system’s behavior. Which an arrange of velocities v = (vx. vy, vz) that are compatible with geography law and can also be given by: B ={(vx, vy, vz) : IR3 → IR3│v falls within the cone-shaped region given} The Roesser Model This model enables the generality of specific analysis structure and design, and one-dimensional state design result space systems by enabling its model’s local state to be divided into a vertical and horizontal state that are propagated respectively, vertically and horizontally by difference equations of the first order[Gop10].

We can find the extension of this model to n-D in Kaczorek (1992) study. With the interest that is growing in the singular system, n-D (2-D) generalization of Roesser models can be done as follows (Kurek, 1989),[Lew92]: Let’s consider hyperbolic partial differential equation With the following initial value of conditions of boundary: T(x, 0) = f1(x) T(0, t) = f2(t) Let If xh (i.j) = T (i – 1.j) and xv (i. j) = T(i, j) is defined, then by rewriting the form Where We will then obtain the Roesser model: Controlled Invariant According to Basile and Marro (1992), the state space controlled invariant presentation for certain systems is considered a subspace in a way that it is possible to control the system in such a manner that all times the state is inside the subspace if the system’s state is initially available in the subspace.

In defining controlled invariant subspace, we will consider a linear system which following differential equation clearly indicates. X(t) = Ax(t) + Bu(t). In which the system’s state and u(t) ∈ Rp is the input assigned by x(t) ∈ Rn The sizes of matrices A and B are n × n and n × p in the same order. If for any x(0) ∈ V, there is an input u(t) in a way that x(t) ∈ V for all non-negative t, then A subspace V ⊂ Rn is considered to be a controlled invariant subspace. According to Gosh, 1985, if and only if AV ⊂ V + Im B will a subspace V ⊂ Rn be considered a controlled invariant (Ntogramatzidis, 2012).

History of Dimensional Spaces In mathematics and physics, bi-dimensional space or two-dimensional space refers to the physical universe's planner projection geometric model. The lengths and widths are the two dimensions, and both directions are lying on the same plane. The 'n' real number sequence is understood as a location in dimensional space. The set of all locations of such kind is referred to as bi-dimensional space or two-dimensional space when n=2. It is usually thought as Euclidean space.

The books of Euclid's elements from I through to IV and VI were majorly concerned with two-dimensional geometry and developed notions as the similarity of shapes, equity of areas and angles, Pythagorean theorem, parallelism, the three cases making the triangle equal, the sum of angles in a triangle and several other topics.

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