# State space representation of electrical system

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Part III introduces dynamic elements and shows how to develop a state-variable model of a dynamic circuit. What’s different from what you learned in physics Modeling of electrical systems is best approached by thinking separately about the element laws and the interaction laws. Example: A State-Space Controller for DC Motor Position Control The electric circuit of the armature and the free body diagram of the rotor are shown in the following figure: For this example, we will assume the following values for the physical parameters. These values were derived by experiment from an actual motor in Carnegie Mellon's • The state of a system is a set of variables such that the knowledge of these variables and the input functions will, with the equations describing the dynamics, provide the future state and output of the system. • For a dynamic system, the state of a system is described in terms of a set of state variables. System u 1(t) u 2(t) y 1(t) y 2(t) In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve through time in a way that depends on the values they have at any given time and also depends on the externally imposed values of input variables. Part III introduces dynamic elements and shows how to develop a state-variable model of a dynamic circuit. What’s different from what you learned in physics Modeling of electrical systems is best approached by thinking separately about the element laws and the interaction laws.

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Difficulties going to State Space from System Diagram Contents. It is often quite easy to develop a state space model from a system diagram (see here for examples). However, there are several situations in which it is not entirely straightforward to develop a state space model from a system diagram. state-space mo dels. DT Mo dels The k ey feature of a state-space description is the follo wing prop ert y, w h i c e shall refer to as the state pr op erty. Giv en presen t v ector (or \state") and input at time t, w e can compute: (i) the presen output, using (7.2); and (ii) next state (7.1). It is easy to see that this puts us in a p osition ... robotics.itee.uq.edu.au 22.451 Dynamic Systems – System Representation State-Space Representation Another approach commonly used is to represent the system as a larger system of equations – The second order ODE with two variables to describe the “state” of the system rather than one variable The number of state variable depends on the robotics.itee.uq.edu.au

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For the electric RLC circuit shown above, the dynamic models will be designated. The first dynamic model will be in form of a transfer function. The second dynamic model will be in form of state space representation equations. Start conditions (initial conditions) for this example are equal to zero (ST=0). The state-space representation can be thought of as a partial reduction of the equation list to a set of simultaneous differential equations rather than to a single higher order differential equation. Although the state variables of a system are not unique and definition of many non-physical variables is possible,

– Model the system using state vector representation – Obtain the state equations • Solve a system of ﬁrst order homogeneous differential equations using state-space method – Identify the exponential solution – Obtain the characteristic equation of the system – Obtain the natural response of the system using eigen-values and vectors The first and the second equations are known as state equation and output equation respectively. The number of the state variables required is equal to the number of the storage elements present in the system. It is a vector, which contains the state variables as elements. In the earlier chapters ... First consider that our uncompensated motor rotates at 0.1 rad/sec in steady state for an input voltage of 1 Volt (this is demonstrated in the DC Motor Speed: System Analysis page where the system's open-loop response is simulated). Since the most basic requirement of a motor is that it should rotate at the desired speed, we will require that ... The state-space representation, also referred to as the time-domain representation, can easily handle multi-input/multi-output (MIMO) systems, systems with non-zero initial conditions, and nonlinear systems via Equation (1). Consequently, the state-space representation is used extensively in "modern" control theory.

the state-space realization topic. Speci cally, an important result will show that if a system can be identi ed by an input-output equation of a particular form, which is fairly general, then a state-space realization can always be easily derived directly from the input-output map. Finally, the theory will be applied to nd a state-space State Space representation of electrical system: Thus, the state of the network at time t=0 is specified by the inductor current and capacitor voltage. Therefore i L (0) and V c (0) is called the initial state of the network and the pair i L (t), V c (t) is called the state of the network at 't'. state-space mo dels. DT Mo dels The k ey feature of a state-space description is the follo wing prop ert y, w h i c e shall refer to as the state pr op erty. Giv en presen t v ector (or \state") and input at time t, w e can compute: (i) the presen output, using (7.2); and (ii) next state (7.1). It is easy to see that this puts us in a p osition ...