Modern Control Engineering

710 Chapter 9 / Control Systems Analysis in State Space Solution.Equation (9–103), where m=4, can be expanded as follows: Since ...

Example Problems and Solutions 711 Solving this last equation for f(A), we obtain wherem=4. Thus, we have shown the equivalence ...

712 Chapter 9 / Control Systems Analysis in State Space In the present case we have (9–106) By substituting l 1 ,l 4 ,p,lmforlin ...

Example Problems and Solutions 713 These msimultaneous equations determine the akvalues (where k=0,1,2,p,m-1). Noting thatf(A)= ...

714 Chapter 9 / Control Systems Analysis in State Space Substitutingl 1 =2, and l 3 =1into these three equations gives Solving f ...

Example Problems and Solutions 715 Since the solution of Equation (9–114) is att=T, we have (9–117) Substituting Equation (9–117 ...

716 Chapter 9 / Control Systems Analysis in State Space Pis equal to that of Q.] If the rank of Pism, then CeATx(0)spans the m-d ...

Example Problems and Solutions 717 The same system may be represented by the following state-space equation, which is in the obs ...

718 Chapter 9 / Control Systems Analysis in State Space If a pole-zero cancellation occurs in the transfer function, then the co ...

Example Problems and Solutions 719 Consequently, for a certain x(0), which implies that, for a certain x(0),x(0)cannot be determ ...

720 Chapter 9 / Control Systems Analysis in State Space Problems B–9–1.Consider the following transfer-function system: Obtain t ...

Problems 721 B–9–11.Obtain a transfer-function representation of the following system with MATLAB: B–9–12.Obtain a transfer-func ...

10 722 Control Systems Design in State Space 10–1 Introduction This chapter discusses state-space design methods based on the po ...

Section 10–2 / Pole Placement 723 systems with observers. Section 10–8 discusses quadratic optimal regulator systems. Note that ...

724 Chapter 10 / Control Systems Design in State Space Consider a control system (10–1) where We shall choose the control signal ...

Section 10–2 / Pole Placement 725 Kis chosen properly, the matrix A-BKcan be made an asymptotically stable matrix, and for all x ...

726 Chapter 10 / Control Systems Design in State Space Notice that the eigenvalues of A 22 do not depend on K. Thus, if the syst ...

Section 10–2 / Pole Placement 727 (10–9) [See Problems A–10–2andA–10–3for the derivation of Equations (10–8) and (10–9).] Equati ...

728 Chapter 10 / Control Systems Design in State Space Now let us simplify the characteristic equation of the system in the cont ...

Section 10–2 / Pole Placement 729 Determination of Matrix K Using Transformation Matrix T. Suppose that the system is defined by ...