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NCSU MAE 208 - SYSTEM CONCEPTS

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SYSTEM CONCEPTS 9 SYSTEM CONCEPTS Automotive vehicles, building structures and aircraft vehicles are dynamical systems but so too are actuators and sensors. Actuators and sensors have their own dynamical behavior, as well. For example, when we say that a motor prescribes a moment, what does that mean? Does it mean that it can apply a moment of any magnitude? Does the moment depend on the inertia of the motor itself? Does the applied moment depend on the physical system attached to it? The short answers to each of these questions are yes. In reality, the actuator and the physical system form a new system that has its own dynamical properties. Most dynamical systems are assemblages of subsystems that are dynamical systems in their own right. The interactions between the subsystems influence the behavior of the overall system. Ideally, it would be much simpler if the dynamical behavior of the subsystems were independent of each other; certainly the system would be much easier to analyze than if we only needed to analyze the dynamic behavior of the subsystems. In fact, this is an important goal of control system design – to put together systems MAE 461: DYNAMICS AND CONTROLSSYSTEM CONCEPTS composed of subsystems that act independent of one another. The principles used to accomplish this design strategy are called separation principles. In this chapter we will first introduce a notation for systems called operator notation. Next, we’ll introduce the block-diagram representation of a system. Then, we introduce the separation principle for tracking and regulation. This is arguably the most important separation principle in control system design. It serves as the basis for what is often referred to as modern controls. I. Linear Operators The previous chapters focused on single degree-of-freedom systems. The single degree-of-freedom system can be represented as (9 – 1) kdtdcdtdmLfLx ++==22 where L is the linear operator of the physical system. In the case of dynamic systems, the linear operator is differential and not merely algebraic. The solution to Eq. (9 – 1) is represented by (9 – 2) fLx1−= where L-1 is the inverse of L. The operator L-1 is symbolic of solving the differential equation. Although the notation does not explicitly deal with the system’s initial conditions, the solution depends on initial conditions so the operation L-1 assumes that the system is acted on by certain initial conditions. MAE 461: DYNAMICS AND CONTROLSSYSTEM CONCEPTS The operator L is said to be linear when the following occurs: Consider the two solutions x1 and x2. The operator L is linear if for any constants α1 and α2 (9 – 3) 22112211)( LxLxxxLαααα+=+ When the operator in Eq. (9 – 1) is linear, Eq. (9 – 1) is said to be a linear equation. The two forces that correspond to the solutions x1 and x2 are and . Letting 11Lxf =22Lxf =2211xxxαα+= it follows from Eq. (9 – 3) that (9 – 4) 2211fffαα+= Equation (9 – 4) expresses the linear superposition principle. In other words, the following three statements are equivalent: (1) An equation is linear. (2) The operator is linear. (3) The linear superposition principle applies. A linear control law can be represented by a linear operator, too. For example, the PID control law is represented below using a linear control gain operator: (9 – 5) ∫++=−= dtidtdhgGxGfRR)( MAE 461: DYNAMICS AND CONTROLSSYSTEM CONCEPTS 2. Block-Diagrams In general, the applied action on a subsystem or a system is called the input and the response of the subsystem or the system is called the output. A convenient way to represent a system consisting of subsystems is the block-diagram representation. Consider, below, the block-diagram representa-tions of a tracking system and a regulation system. Figure 9 – 1 Tracking System Fig 9 – 2 Regulation System Figure 9 – 2 Regulation System First consider the regulation system. In Fig. 9 – 1, we see two blocks; one represents the controller GR, and the other represents the plant L. The plant could be the single degree-of-freedom system MAE 461: DYNAMICS AND CONTROLSSYSTEM CONCEPTS described in Eq. (9 – 1) and GR could represent the PID feedback controller given in Eq. (9 – 5). Also, notice that the quantities that enter a given node sum to zero. A minus sign next to a quantity entering a node indicates that the negative of that quantity enters the node. The quantity r(t) that enters the left node is called the reference input, and the quantity is called the disturbance input. Notice that the equations that are represented in the block-diagram of the regulation system are: )(tfD (9 – 6) fLxffffxrGdtidtdhgGkdtdcdtdmLDRRRR122)()(−=+==−++=++=∫ The reference input in the block-diagram of the regulation system represents the reference point (equilibrium position) about which the motion is controlled. Next, consider the block-diagram of a tracking system shown in Fig. 9 - 2. Here, the reference input r(t) represents the desirable path specified by the control system designer. The left block GT is the tracking controller, the right block is the plant L and is the disturbance input. Again, the plant could be the single degree-of-freedom system described in Eq. (9 - 1) and GT could represent a tracking controller whose output attempts to mimic the reference input. The list of equations that are represented in the block-diagram of the tracking control system is: )(tfD MAE 461: DYNAMICS AND CONTROLSSYSTEM CONCEPTS (9 – 7) fLxfffrGfkdtdcdtdmGkdtdcdtdmLDTTTT10022022−=+==++=++= In Eq. (9 – 7), m, c, and k are the physical parameters of the plant, while are the postulated physical parameters of the tracking controller. The postulated physical parameters are selected to be as close as possible to the physical parameters. 00,0and , k cm 3. Separation Principle for Tracking and Regulation The question arises how to combine tracking and regulation. The block-diagram shown in Fig. 9 - 3 combines tracking and regulation while maintaining a separation principle between them. This is the block-diagram most frequently used in modern control systems. Fig. 9 - 3 Figure 9 – 3 Combined Tracking and Regulation System MAE 461: DYNAMICS AND CONTROLSSYSTEM CONCEPTS The equations that are represented in the block-diagram for the combined tracking


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