Systems Engineering For Dummies
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Using the result from our example, we can condense the left side of our equation into a derivative: Now we can integrate both sides, from the initial time ( t 0) to the current time ( t), using a dummy variable τ, we will get closer to our result. Finally, if we premultiply by e At, we get our final result: [General State Equation Solution] If we plug this solution into the output equation, we get: [General Output Equation Solution] This is the general Time-Invariant solution to the state space equations, with non-zero input. These equations are important results, and students who are interested in a further study of control systems would do well to memorize these equations. State-Transition Matrix [ edit] More information about matrix exponentials can be found in: Engineering Analysis The state transition matrix, e At, is an important part of the general state-space solutions for the time-invariant cases listed above. Calculating this matrix exponential function is one of the very first things that should be done when analyzing a new system, and the results of that calculation will tell important information about the system in question.
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State Equation Solutions [ edit] The solutions in this chapter are heavily rooted in prior knowledge of Ordinary Differential Equations. Readers should have a prior knowledge of that subject before reading this chapter. The state equation is a first-order linear differential equation, or (more precisely) a system of linear differential equations. Because this is a first-order equation, we can use results from Ordinary Differential Equations to find a general solution to the equation in terms of the state-variable x. Once the state equation has been solved for x, that solution can be plugged into the output equation. The resulting equation will show the direct relationship between the system input and the system output, without the need to account explicitly for the internal state of the system. The sections in this chapter will discuss the solutions to the state-space equations, starting with the easiest case (Time-invariant, no input), and ending with the most difficult case (Time-variant systems).
Systems engineering for dummies
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