state variables help 1

state variables help 1 - EE 422G Notes Chapter 7 Instructor...

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EE 422G Notes: Chapter 7 Instructor: Cheung Page 7-5 Use State Variable to represent circuits Let’s start with a simple example: NOTE: in this chapter, we will use u(t) (not to be confused with the step function) to denote the input and use x(t) to denote the state. 1. Define the states: Similar to the discrete-case, we define the states based on the memory storage elements. For passive circuit, the memory storage elements are the capacitors and inductors: dt di L v and dt dv C i L L C C = = As the knowledge capacitor voltage and inductor current allows us to infer the capacitor current and the inductor voltage by taking derivatives, we define state variables to be the capacitor voltage and inductor current . ) ( 1 t v x c = ) ( 2 t i x L = Note that there is one state variable for each memory storage element . 2. Derive the “State” and “Output” equations – express the derivatives of the states and the output in terms of the current state and the input ONLY. KCL at node a: 2 1 1 x C dt dx = KVL: u L x L R x L dt dx 1 1 2 1 2 + - - = These two equations can be more compactly written in matrix form: u L x x L R L C x x + - - = / 1 0 / / 1 / 1 0 2 1 2 1 & & The above matrix equation is called the State Equation because it relates the change (1 st -order derivative) of the state to the current state and input. We can also relate the output to the state variables as follows: Output Equation : ( 29 = 2 1 0 1 ) ( x x t y R=2 L=1H + - u(t) + - y(t) C=1F a i

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EE 422G Notes: Chapter 7 Instructor: Cheung Page 7-6 Using this approach, we can write the state-variable representation for any circuit. In general, the state and output equations are always in the following form: In our previous example, we have ( 29 0 , 0 1 , / 1 0 , / / 1 / 1 0 = = = - - = D C L B L R L C A Be very careful about the dimensions of each matrix. Let’s do another example:
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