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StateSpace - 2.14 Analysis and Design of Feedback Control...

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2.14 Analysis and Design of Feedback Control Systems State-Space Representation of LTI Systems Derek Rowell October 2002 1 Introduction The classical control theory and methods (such as root locus) that we have been using in class to date are based on a simple input-output description of the plant, usually expressed as a transfer function. These methods do not use any knowledge of the interior structure of the plant, and limit us to single-input single-output (SISO) systems, and as we have seen allows only limited control of the closed-loop behavior when feedback control is used. Modern control theory solves many of the limitations by using a much “richer” description of the plant dynamics. The so-called state-space description provide the dynamics as a set of coupled first-order differential equations in a set of internal variables known as state variables , together with a set of algebraic equations that combine the state variables into physical output variables. 1.1 Definition of System State The concept of the state of a dynamic system refers to a minimum set of variables, known as state variables , that fully describe the system and its response to any given set of inputs [1-3]. In particular a state-determined system model has the characteristic that: A mathematical description of the system in terms of a minimum set of variables x i ( t ), i = 1 , . . . , n , together with knowledge of those variables at an initial time t 0 and the system inputs for time t t 0 , are sufficient to predict the future system state and outputs for all time t > t 0 . This definition asserts that the dynamic behavior of a state-determined system is completely characterized by the response of the set of n variables x i ( t ), where the number n is defined to be the order of the system. The system shown in Fig. 1 has two inputs u 1 ( t ) and u 2 ( t ), and four output vari- ables y 1 ( t ) , . . . , y 4 ( t ). If the system is state-determined, knowledge of its state variables ( x 1 ( t 0 ) , x 2 ( t 0 ) , . . . , x n ( t 0 )) at some initial time t 0 , and the inputs u 1 ( t ) and u 2 ( t ) for t t 0 is sufficient to determine all future behavior of the system. The state variables are an internal description of the system which completely characterize the system state at any time t , and from which any output variables y i ( t ) may be computed. Large classes of engineering, biological, social and economic systems may be represented by state-determined system models. System models constructed with the pure and ideal (linear) one-port elements (such as mass, spring and damper elements) are state-determined 1
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Figure 1: System inputs and outputs. system models. For such systems the number of state variables, n , is equal to the number of independent energy storage elements in the system. The values of the state variables at any time t specify the energy of each energy storage element within the system and therefore the total system energy, and the time derivatives of the state variables determine the rate of change of the system energy. Furthermore, the values of the system state variables at any time t
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