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lecture8_small - State-space systems Approach: Represent...

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State-space systems Representations: From transfer function to state-space. Consider a linear, shift invariant, system: P ( z ) ±± u ( k ) y ( k ) We can express this as a transfer function, y ( z )= P ( z ) u ( z b ( z ) a ( z ) u ( z ) where a ( z ) and b ( z ) are polynomials, so, P ( z b ( z ) a ( z ) = b 0 z m + b 1 z m - 1 + ··· + b m z n + a 1 z n - 1 + + a n . For causal systems the order of b ( z ) is less than or equal to the order of a ( z ). So m n above. Assume for now that m < n , Roy Smith: ECE 147b 8 :2 State-space systems Approach: Represent the plant, P ( z ) (or P ( s )) as an n th order di±erential equation. Represent n th order di±erential equation as a 1st order matrix di±erential equation with dimension n . Design methods now involve linear algebra. Easy to handle large systems (with Matlab ). Easy to handle systems with multiple inputs and outputs. Easy to simulate systems. x ( k + 1) = Ax ( k )+ Bu ( k ) , y ( k Cx ( k Du ( k ) . A , B , C and D can be matrices. x ( k ) is a vector (state vector). Roy Smith: ECE 147b 8 :1
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State-space systems Chain of delays: Consider the frst equation: ζ ( z )= 1 a ( z ) u ( z ) We want to develop a chain oF delay model to get ζ ( k ): First step: Write the expression For n delays: ζ ( k z - 1 ζ ( k + 1) . . . . . . ζ ( k + n - 1) = z - 1 ζ ( k + n ) In pictures . .. z - 1 z - 1 z - 1 ± ± ± ± ± ζ ( k + n ) ζ ( k + 1) ζ ( k + n - 2) ζ ( k + n - 1) ζ ( k ) Second step: Express ζ ( k + n ) in terms oF ζ ( k ), . . . , ζ ( k + n - 1) and u ( k ). Roy Smith: ECE 147b 8 :4 State-space systems Outline: 1. Draw the system as an interconnected “chain oF delays”, 2. Relabel the signals in the system, 3. Rewrite the input/output equations in terms oF the new signals, 4. Abbreviate the equations to a matrix Form (state-space). Drawing a digital system block diagram in terms oF delays is exactly the same as drawing a continuous system block diagram in terms oF integrators. Split the system, b ( z ) 1 a ( z ) ± ± ± u ( z ) ζ ( z ) y ( z ) ζ ( z 1 a ( z ) u ( z ) and y ( z b ( z ) ζ ( z ) .
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lecture8_small - State-space systems Approach: Represent...

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