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Note for Advanced Control Systems - ACS by mahaveer singh

  • Advanced Control Systems - ACS
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  • Electrical Engineering
  • B.Tech
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Chapter 3 State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal-Flow Graph State Variables The Transfer Function from the State Equation 1

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Introduction • In the previous chapter, we used Laplace transform to obtain the transfer function models representing linear, time-invariant, physical systems utilizing block diagrams to interconnect systems. • In Chapter 3, we turn to an alternative method of system modeling using time-domain methods. • In Chapter 3, we will consider physical systems described by an nth-order ordinary differential equations. • Utilizing a set of variables known as state variables, we can obtain a set of first-order differential equations. • The time-domain state variable model lends itself easily to computer solution and analysis. 2

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Time-Varying Control System • With the ready availability of digital computers, it is convenient to consider the time-domain formulation of the equations representing control systems. • The time-domain is the mathematical domain that incorporates the response and description of a system in terms of time t. • The time-domain techniques can be utilized for nonlinear, timevarying, and multivariable systems (a system with several input and output signals). • A time-varying control system is a system for which one or more of the parameters of the system may vary as a function of time. • For example, the mass of a missile varies as a function of time as the fuel is expended during flight 3

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Terms • • • • • State: The state of a dynamic system is the smallest set of variables (called state variables) so that the knowledge of these variables at t = t0, together with the knowledge of the input for t ≥ t0, determines the behavior of the system for any time t ≥ t0. State Variables: The state variables of a dynamic system are the variables making up the smallest set of variables that determine the state of the dynamic system. State Vector: If n state variables are needed to describe the behavior of a given system, then the n state variables can be considered the n components of a vector x. Such vector is called a state vector. State Space: The n-dimensional space whose coordinates axes consist of the x1 axis, x2 axis, .., xn axis, where x1, x2, .., xn are state variables, is called a state space. State-Space Equations: In state-space analysis, we are concerned with three types of variables that are involved in the modeling of dynamic system: input variables, output variables, and state variables. 4

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