State space trajectory

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Phase space

The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position and momentum parameters. It is the direct product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.

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• State Space Representation of Dynamical Systems

  | | | (10) | Instantaneous state of a dynamic system and other behavior considered as clearly defined if at some point in time t=ti the known value of the n state variables xi (i = 1, n). These values can be considered as the coordinates of the point (x1, x2, ..., xn) of n-dimensional space, which is called the state space. Point with coordinates x1,x2...xn representing a curve point, after which point to the change of status call state trajectory. The state trajectory in three-dimensional state space is shown in Figure 1. Download FigureDownload as PPT PowerPoint Slide PNG Larger image(png format) View FigureVeiw figureFigures index NEW View larger figure in new window View next figure Figure 1. Determining the notion of the state trajectory [...] Figure 1. Determining the notion of the state trajectory To specific initial conditions x1(0), x2(0),...xn(0) corresponds to one solution of the dynamic system. This means that each group of initial conditions corresponding to only one start point M0 and one status trajectory. Corresponds to the set of group of initial conditions corresponding to the whole set of state trajectory, which we call the phase portrait of a dynamical system. The state trajectory set really gives you an idea about the behavior of dynamic systems in time. State-space method is best suited for the analysis of a dynamic system of the second order, because in the this case the state trajectory in the plane s. [...] This method is simple and opinions shall form the dynamic system of 2nd order. In this case, the state space is represented by a plane state trajectories are in the plane of the curve. State trajectory can be obtained directly from the equation of state without having to look for their solutions. State trajectory equation so we can get and we do not know the given system of equations of state directly addressed. Suppose dynamic 2nd order system is described by nonlinear differential equations: | | | (11) | Substitution: | | | (12) | Then we get a system of two equations of state: | | | (13) | If in the system (13) the second equation divided by the first equation by we get: | | | (14) |

• State Trajectory - an overview

  State Trajectory. In subject area: Engineering. State trajectory is defined as the path that a system's state variables follow over time as it evolves

• LINEAR STATE-SPACE CONTROL SYSTEMS

  Specifically, we derive conditions under which, starting anywhere in the state space, the state trajectory can be driven to the origin by piecewise continuous input signals over a finite time interval. More generally, it turns out that if it is possible to do this, then it is also possible to steer the state trajectory to any final state in finite time via a suitable input signal. While the controllability analysis presented in this chapter has a decid-edly open-loop flavor involving input signals that have a prescribed effect on the state trajectory without any mention of feedback, there is also an important connection between controllability and the design of feedback-control laws that will be pursued in Chapter 7. [...] Definition 1.1 For a nominal input signal, ˜ u(t), the nominal state tra-jectory ˜ x(t) satisfies ˙ ˜ x(t) = f [˜ x(t), ˜ u(t), t] and the nominal output trajectory ˜ y(t) satisfies ˜ y(t) = h[˜ x(t), ˜ u(t), t] If ˜ u(t) = ˜ u, a constant vector, a special case is an equilibrium state ˜ x that satisfies 0 = f (˜ x, ˜ u, t) for all t. [...] For positive damping (c > 0), we have dE/dt < 0 along any trajectory for which the mass velocity is not identically zero. A trajectory for which x2(t) = ˙ y(t) is identically zero corresponds to identically zero accelera-tion and a constant displacement. Since such a trajectory also must satisfy the equations of motion we see, on substituting ˙ x2(t) = ¨ y(t) ≡0, ˙ x1(t) = x2(t) = ˙ y(t) ≡0, and x1(t) = y(t) ≡y0 (constant), that ky0 = 0. Conse-quently, the only trajectory for which the mass velocity is identically zero corresponds to the equilibrium state x(t) ≡˜ x = [0, 0]T . We conclude that the total energy in the system is strictly decreasing along all other tra-jectories and converges to zero as time tends to infinity. We expect that this should correspond to asymptotic convergence of

• Change in State Space: from time series to trajectories and back again | Modeling Life

  20 Chaos and Cardiology 20.1 Cardiac Chaos20.2 Periodically Stimulated Cardiac Cells20.3 Ventricular Fibrillation 21 Applications of Linear Algebra 21.1 Applications of Linear Algebra 1 - WWW2 Strategic Bombing21.2 Applications of Linear Algebra 2 - The Google PageRank Algorithm21.3 Applications of Linear Algebra 3 - Google PageRank and Foodwebs 22 Interactive Simulations 22.1 A Model of HIV Immunology within an Individual Person22.2 Grass-Sheep-Wolves22.3 Shark-Tuna22.4 Lac Operon22.5 Epidemic modeling # Change in State Space: from time series to trajectories and back again ## Summary What is “change”? + change is a curve through state space, called a trajectory. Going back and forth between state space trajectories and time series [...] Modeling LifeInteractive SimulationsUnderstanding DataVideos 1 State Space 1.1 Introduction to State Variables and State Space1.2 Defining Vectors: Working With n-Dimensional Space 2 Operating with Vectors 2.1 State Space: Rules for Operating with Vectors2.2 Addition and Scalar Multiplication of 2-Dimensional Vectors 3 Trajectories and Time Series 3.1 Vectors and Trajectories - a little history3.2 Change in State Space: from time series to trajectories and back again3.3 Functions 4 Models and Change Vectors [...] 4 Models and Change Vectors 4.1 Introduction to Models and Change Equations: The Bathtub4.2 Models and Change Equations in 2 Dimensions: Shark Meets Tuna4.3 From Change Equations to Change Vectors4.4 State spaces and change vectors in 2D4.5 Trajectories in 2D4.6 Logistic Growth4.7 Models and Vectorfields: a nonlinear example in 1D4.8 Modeling Chemical Reactions4.9 Epidemiology4.10 Immune Dynamics 5 Euler's Method 5.1 Introducing Euler's Method5.2 Algebra of Euler's Method in 1D5.3 Algebra of Euler's Method in 2D5.4 Katherine Johnson and Euler's Method5.5 Euler's Method to Create a Model of Ventricular Fibrillation 6 Equilibrium Points 6.1 Equilibrium Points in 1D6.2 Stable and Unstable Equilibrim Points 7 Linear Stability Analysis

• State space - Scholarpedia

  Of course, not every arbitrarily drawn curve in the phase plane represents a solution. What is special about solution trajectories is that the velocity vector at each point along the trajectory is given by the right hand side of the differential equation above. That is, the velocity vector of the trajectory \((x(t), y(t))\) at a point \((x(p), y(p))\) is given by \((x', y') = (f(x(p),y(p)), g(x(p),y(p)))\ .\) This geometric property -- that the vector \((f(x,y), g(x,y))\) always points in the direction that the solution is flowing -- completely characterizes the solution trajectories (considered as subsets of the phase space). The function that assigns \((f(x,y), g(x,y))\) to \((x, y)\) is called the vector field. [...] ## Higher dimensional systems More generally, consider a system of \(n\)-first order differential equations of the form: \[ u' = F(u), ~~~ u \in R^n\ .\] The phase space is simply \(n\)-dimensional Euclidean space") and every solution, \(u(t)\ ,\) corresponds to a trajectory in phase space parametrized by the independent variable \(t\ .\) As before, \(F(u)\) defines a vector field in the phase space; at each point, \(u(p)\ ,\) the vector \(F(u(p))\) must be tangent to the solution curve \(u(t)\ .\) Moreover, equilibria are where \(F(u)=0\) and periodic solutions correspond to closed orbits. [...] Phase planes typically arise in the context of two-dimensional autonomous ODEs, which can be written in the form \[x' = f(x,y)\] \[y' = g(x,y)\ .\] Here, \(f\) and \(g\) are given (smooth) functions. The two variables could describe, e.g., the position and velocity of a particle, the state of a predator-prey system, or concentrations of two reagents in a homogeneous chemical reaction. If \((x(t), y(t))\) is a solution of the system, then at each time \(t=p\ ,\) \((x(p), y(p))\) defines a point in the phase plane. The point changes with time, so the entire solution, \((x(t),y(t))\ ,\) traces out a curve, or trajectory, in the phase plane.