GPS with High Rate Sensors

State Space Linearization (GPS)

Navigation system analysis, and portions of the implementation, involve linearization of a system about a nominal trajectory. This section explains the linearization process. Let the state space model for a system with input u and output y be described by Assume that for a nominal inputa nominal state trajectoryis known which satisfies Define the error […]

Discrete-Time State Space Notation (GPS)

It is often advantageous to work with discrete-time equivalent models of continuous-time systems, particularly with the advent of the digital computer. This section discusses only notation definition for state space models of discrete-time systems. The calculation of discrete-time models equivalent to continuous-time systems is discussed in Section 3.5.5. The standard form for a time-invariant, discrete-time, […]

State Space Analysis (GPS) Part 1

The previous sections have casually referred to the state of a system. The following definition formalizes this concept. Definition 3.1 The state of a dynamic system is a set of real variables such that knowledge of these variables at time to together with knowledge of the system input foris sufficient to determine the system response […]

State Space Analysis (GPS) Part 2

State Transition Matrix Properties The homogeneous part of eqn. (3.43) is Definition 3.2 A continuous and differentiable matrix function is the fundamental solution of eqn. (3.53) onif and only iffor all The fundamental solutionis important since it will serve as the basis for finding the solution to both eqns. (3.43) and (3.53). Ifthen x(t) satisfies […]

State Estimation (GPS) Part 1

The state space model format clearly shows that a system may have many internal variables (e.g., states) and fewer outputs (e.g., y). This is true for a variety of reasons including cost, power, or lack of appropriate sensors. If knowledge of the state vector is desired, but is not directly measured, then we have the […]

State Estimation (GPS) Part 2

Estimator Design by Pole Placement This section discusses estimator design by pole placement (i.e., Ackerman’s method) for two reasons.Second, understanding of the concepts of this section will aid the understand of the results derived in Section 3.6.3. Section 3.1.3 derived the controllable canonical form state space representation for the strictly proper transfer function in eqn. […]

State Estimation (GPS) Part 3

Observable Subspace Let X denote the state space of a system. When the system is not observable (for a given set of sensors), it is possible to define a similarity transform,such that the vector v can be partitioned into a set of stateswhich are observable and a set of stateswhich are unobservable: whereare partitioning matrices […]

Basic Stochastic Process Concepts (GPS)

Stochastic Processes Navigation systems combine uncertain information related to a vehicle to construct an accurate estimate of the vehicle state. The previous topic discussed deterministic models of dynamic systems that are appropriate for modeling the vehicle dynamics or kinematics. In navigation applications, the information uncertainty does not typically arise from the vehicle dynamics; instead, it […]

Scalar Random Variables (GPS)

This section presents various fundamental concepts related to probability and random variables that will be required to support subsequent discussions. The discussion will focus on continuous random variables. Basic Properties Let w represent a real, scalar random variable. In the following, the notationdenotes the probability that the random variable w is less than the real […]

Multiple Random Variables (GPS)

Navigation systems frequently involve multiple random variables. For example, a vector of simultaneous measurements y might be modeled as where y represents the signal portion of the measurement and v represents a vector of random measurement errors. In the case of multiple random variables, we are often concerned with how the values of the elements […]