Chapter1_S12 - Chapter 1 Linear Dynamical Systems 1.1...

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Chapter 1 Linear Dynamical Systems 1.1 System classifications and descriptions A system is a collection of elements that interacts with its environment via a set of input variables u and output variables y . Systems can be classified in di ff erent ways. Continuous time versus Discrete time A continuous time system evolves with time indices t , whereas a discrete time system evolves with time indices t Z = { . . . , 3 , 2 , 1 , 0 , 1 , 2 , . . . } . Usually, the symbol k instead of t is used to denote discrete time indices. Examples of continuous time systems are physical systems such as pendulums, servomechanisms, etc. An examples of a discrete time system is mutual funds whose valuation is done once a day. An important class of systems are sampled data systems. The use of modern digital computers in data processing and computations means that data from a continuous time system is “sampled” at regular time intervals. Thus, a continuous time system appears as a discrete time system to the controller (computer) due to the sampling of the system outputs. If inter-sample behaviors are essential, a sampled data system should be analyzed as a continuous time system. Static versus dynamic The system is a static system if its output depends only on its present input. i.e. (there exists) a function f ( u, t ) such that for all t T , y ( t ) = f ( u ( t ) , t ) . (1.1) An example of a static system is a simple light switch in which the switch position determines if the light is on or not. A static time invariant system is one with y ( t ) = f ( u ( t )) for all t . To determine the output of a static system at any time t , the input value only at t is needed. Again, a light switch is a static time invariant system. A static time-varying system is one with time-varying parameters such as external disturbance signals. An example is a flow control valve (Fig.1.1), whose output flow rate Q is given as Q = uA 2 P s ( t ) ρ (1.2) 3
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4 c Perry Y.Li A uA Q P Figure 1.1: Flow control valve: a static time invariant system A uA Q Fs x P Figure 1.2: Flow control valve: a dynamic time invariant system where u [0 , 1] is the input, A is orifice area, P s is flow pressure and ρ is fluid density. Here P s is a time-varying parameter that a ff ects the static output Q . In contrast, a (causal) dynamic system requires past input to determine the system output. i.e. to determine y ( t ) one needs to know u ( τ ), τ ( −∞ , t ]. An example of a dynamic time invariant system is the flow control valve shown in Fig. 1.2. The fluid pressure P s is constant. However, the flow rate history is a function of the force F ( t ) acting on the valve. It is necessary to know the time history of the forcing function F ( t ) in order to determine the flow rate at any time. The position x ( t ) of the valve is governed by the di ff erential equation ¨ x = F ( t ) b ˙ x kx (1.3) where k is the spring constant and b is the damping factor. For a circular pipe of radius R , the flow rate is then given by: Q = x 2 R 2 A 2 P s ( t ) ρ (1.4)
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University of Minnesota ME 8281: Advanced Control Systems Design, 2001-2012 5 Earth Orbiting pendulum Figure 1.3: Example of a dynamic time varying system
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