Modern Control Engineering

690 Chapter 9 / Control Systems Analysis in State Space which may be rewritten as (9–74) Noting that we can rewrite Equation (9– ...

Example Problems and Solutions 691 A–9–2. Consider the following transfer-function system: (9–77) Derive the following observabl ...

692 Chapter 9 / Control Systems Analysis in State Space Then Equation (9–80) can be written as (9–82) By substituting Equation ( ...

Example Problems and Solutions 693 A–9–3. Consider the transfer-function system defined by (9–83) where Derive the state-space r ...

694 Chapter 9 / Control Systems Analysis in State Space The inverse Laplace transforms of these equations give (9–87) These nequ ...

Example Problems and Solutions 695 then we get a slightly different state-space representation. This choice of state variables g ...

696 Chapter 9 / Control Systems Analysis in State Space Solution.The partial-fraction expansion of Equation (9–91) becomes which ...

Example Problems and Solutions 697 The inverse Laplace transforms of the preceding nequations give The output equation, Equation ...

698 Chapter 9 / Control Systems Analysis in State Space u y s+^1 p c^1 1 1 s+p 1 1 s+p 1 s+^1 p c^4 4 c 2 c 3 x 3 x 4 xn x 2 x 1 ...

Example Problems and Solutions 699 This is the MATLAB representation of the following state-space equations: A–9–6. Consider the ...

700 Chapter 9 / Control Systems Analysis in State Space wherekis a vector whose components are the magnitudes of rstep functions ...

Example Problems and Solutions 701 Since we have Sincex(0)= 0 andk=1,referring to Equation (9–96), we have Hence, the output y(t ...

702 Chapter 9 / Control Systems Analysis in State Space where the greatest common divisor of the n^2 elements (which are functio ...

Example Problems and Solutions 703 A–9–9. If an n*nmatrixAhasndistinct eigenvalues, then the minimal polynomial of Ais identical ...

704 Chapter 9 / Control Systems Analysis in State Space but Thus, we have shown that the minimal polynomial and the characterist ...

Example Problems and Solutions 705 A–9–10. Show that by use of the minimal polynomial, the inverse of a nonsingular matrix Acan ...

706 Chapter 9 / Control Systems Analysis in State Space By identifying the coefficients aiof the minimal polynomial (which is th ...

Example Problems and Solutions 707 Noting that x(t)can also be given by the equation we obtain or (9–101) Next, we shall conside ...

708 Chapter 9 / Control Systems Analysis in State Space Similarly, if then A–9–12. Consider the following polynomial in lof degr ...

Example Problems and Solutions 709 Assuming that the eigenvalues of an n*nmatrixAare distinct, substitute Aforlin the polynomial ...