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Digital Signal Processing © Dr. Fred J. Taylor, Professor STATE_Determined_SYS - 1 Lesson Title: State Determined Systems Lesson Number [17] / Lecture [19] What’s it all about? ã How do state variables support the analysis of linear systems? ã What is a state transition matrix? State Variable Model The single-input single-output Nth order LTI system shown in Figure 1 can be modeled in terms of a state 4-tuple [A, b, c, d] which defines an N th order state variable model:
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  Digital Signal Processing  Dr. Fred J. Taylor, Professor   STATE_Determined_SYS - 1 Lesson Title: State Determined SystemsLesson Number [17] / Lecture [19] What’s it all about? ã How do state variables support the analysis of linear systems? ã What is a state transition matrix? State Variable Model The single-input single-output N  th order LTI system shown in Figure 1 can be modeled in termsof a state 4-tuple [ A , b , c , d] which defines an N  th order state variable model:State equation:][][]1[ k u k k  bAxx +=+ 1Initial condition:   0 ]0[ xx = 2Output equation: ][][][ k du k k  T  += xcy . 3where  x  [ k  ] is the state n -vector, u  [ k  ] is the input, y  [ k  ] is the output,  A is an n x n state matrix, b isan 1x n input vector, c  is a 1xn output vector, and d is a scalar. At some sample index k  = k  0 , thenext state can be computed to be:][][]1[ 000 k u k k  bAxx +=+ , 4At sample instant k  = k  0 +1, the next state is given by:].1[][][]1+[]1[]2[ 0002000 +++=++=+ k u k u k k u k k  bAbxAbAxx 5Continuing this pattern, at sample k  = k  0 + n  –1 the next state vector is given by:].1+[]2[]1+[][][][ 00020100 −+−+++++=+ −− nk u nk u k u k u k nk  nnn bAbbAbAxAx L 6Equation 6 consists of two parts that define the homogeneous and inhomogeneous solutions.The homogeneous solution is driven only by the initial conditions( x [ k  0 ]), whereas theinhomogeneous solution is forced by the input u  [ k  ]. Equation 6 can be rearranged andexpressed as: ∑ −=−−− += 110 00 ][][][ k k i i k k k  i u k k  bAxAx . 7  Digital Signal Processing  Dr. Fred J. Taylor, Professor   STATE_Determined_SYS - 2Upon substituting Equation 7 into the output equation (Equation 3), the generalized output of thestate determined system is:     ++=+= ∑ −=−−− 110 00 ][][][][][][ k k i i k T k k T T  k du k u k k du k k  bAcxAcxcy . 8For notational convenience, it shall be assumed that k  0 =0. The term ΦΦΦΦ [ k  ]= A k  , is called the statetransition matrix and is seen to influence both the homogenous and inhomogeneous solutions.After simplification, Equation 8 can be recast as: [ ] [ ] k u d i k k k  k i T T  [1]0[][ 10     +−−Φ+Φ= ∑ −= bcxcy . 9The state transition matrix can be computed using Cayley-Hamilton method, z  -transform withthe later normally implemented using Leverrier’s algorithm. It is generally accepted, however,that computing y[k] using time-domain convolution methods, as described in Equation 9, iscomputationally intensive. Instead, transform-domain techniques are often championed.Observe that the   z  -transform of Equations 1 and 3 are: ( ) ( ) ( )( ) ( ) ( ) z dU z z  z U z zx z z  T  ==+=− Xc YbAXX 0 10The homogenous response can be determined by setting U  ( z  )=0 and solving for  Y  ( z  ) inEquation 10. This results in: ( )( ) 001001 )()( ;)()( xcxAIc Y xxAIX z z z z  z z z z  T T  Φ=−= Φ=−= −− . 11This can be seen to be consistent in form with the results given in Equation 9. Theinhomogeneous solution can be computed by setting x 0 = 0 and solving for  Y  ( z  ) using:).()()()( )()()( 11 z dU z U z z Y  z U z z  T  +−=−= −− bAIc bAIX 12The system’s transfer function is associated with the inhomogeneous solution of an at-rest   system, and is defined by H  ( z  )= Y  ( z  )/ U  ( z  ). It therefore follows that: d z z H  T  +−= − bAIc 1 )()(. 13 Ab c T  Tdu[k] Σ x[k+1]x[k]y[k] Σ Figure 1: State Determined System  Digital Signal Processing  Dr. Fred J. Taylor, Professor   STATE_Determined_SYS - 3Notice the similarity between the homogeneous solution (Equation 11) and the inhomogeneoussolution (Equation 13). The homogeneous response is defined in terms of the state transitionmatrix ( ) 1 )( − −=Φ AI z z z  while the inhomogeneous solution is defined in terms of convolutionwith a term having the form 1 )( − − AI z  in the transform domain. Notice that ( ) z z z z z z  Φ=−=− −−−− 1111 )(()( AIAI or matrix transition matrix delayed by one sample in thetime domain. Example: Transfer Function An at-rest 2 nd order linear time-invariant (LTI) system is assumed to be given by the state 4-tuple ( A , b , c , d):1;11;10;1110 ==== d  cbA .Computing ( z  I  – A )  –1 : ( ) −−−=− − z z z z z  11111 21 AI .Using Equation 10, the transfer function can then be determined and it is: [ ] .11111101111111=)()( 22221 −−=+−−+=+−−−+−= − z z z z z z z z z z d z z H  T  bAIc   Example: State Determined Systems The response of a linear system, characterized by the state 4-tuple [ A , b , c , d], is modeled as: [ ] [ ] i bu k  k i i k  ∑ −=−− = 101 Ax   [ ] [ ] ][ 101 k du i bu k  k i i k T  += ∑ −=−− Acy  In the z-domain. Tthe solution can be described as: ( ) )()( 1 z U zI z  bAX − −=   ( ) ( ) ( ) ( ) ( ) ( ) z U z H z U d zI z dU z U zI z  T T  =+−=+−= −− bAcbAc Y 11 )()()( Suppose a specific 2 nd order system is given by: [ ] 1;011.0;10;111.0 10 ==== d c b A  The architecture associated with this manifestation [ A , b , c ,d] is shown in Figure 2.  Digital Signal Processing  Dr. Fred J. Taylor, Professor   STATE_Determined_SYS - 4Figure 2: State induced architectureThe location of the system’s poles are defined by ∆ ( z  )=det( z  I - A )=det( −−− 111.0 1 z z  )= z  2 - z  -0.11=( z  +0.1)( z  -1.1)Therefore the filter's poles are located at z=-0.1 and z=1.1. The system is therefore seen to beunstable based on the unit circle criterion. Continuing, the state transition matrix denoted Φ (z) inthe z-domain, is given by: ( )( ) ( )( ) ( )( )( )( ) −−+=−=ΦΦΦΦ=Φ − z z z z z  AzI z z z z z z  11.0111.11.0 22 1211211  Using Heaviside's method to inverting each term (4 required), the state transition in the time-domain is obtained for k ≥ 0, and it is: [ ] ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) [ ] I k  A k k k k  k k k k  k  ==Φ+−+−− +−−+− =Φ= 10010;1.112111.01211.1121.11.0121.11.112101.012101.11211.01211 The instability of the system can be immediately seen by the presence of the term (1.1) k .Suppose the system is initially at-rest  and the input to the at-rest system is u  [ k  ]= δ   [ k  ]. Then  X  ( z  )= ( zI  -  A ) -1 bU  ( z  )= ( zI  -  A ) -1 b and it follows that: ( ) ( )( )( ) ( ) ( )( )( )( ) [ ][ ] 00001111.11.0 1011.0111.11.0 1 221222112111 >=−Φ−Φ↔ΦΦ=−+==−−+=−= −−−− k if k if k k z z z z z z z z z z z z z b AzI z  X  Z   For example, direct state computation at k  =1 and 2, yields:u[k]b=1 Σ x 2 [k+1]x 2 [k]x 1 [k+1]x 1 [k] Σ a 21 =0.11y[k]c 1 =0.11d=1 Σ a 22 =1a 12 =1   TT
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