Modern Control Engineering

650 Chapter 9 / Control Systems Analysis in State Space (9–4) The controllable canonical form is important in discussing the pol ...

(9–8) (9–9) Jordan Canonical Form. Next we shall consider the case where the denominator polynomial of Equation (9–2) involves m ...

652 Chapter 9 / Control Systems Analysis in State Space EXAMPLE 9–1 Consider the system given by Obtain state-space representati ...

Section 9–2 / State-Space Representations of Transfer-Function Systems 653 (9–12) the transformation x=Pz, where P= l 1 ,l 2 ,p, ...

654 Chapter 9 / Control Systems Analysis in State Space EXAMPLE 9–2 Consider the following state-space representation of a syste ...

Section 9–2 / State-Space Representations of Transfer-Function Systems 655 Simplifying gives (9–20) Equation (9–20) is also a st ...

656 Chapter 9 / Control Systems Analysis in State Space Then we may take as another set of state variables any set of functions ...

Section 9–3 / Transformation of System Models with MATLAB 657 Another possible state-space representation (among infinitely many ...

658 Chapter 9 / Control Systems Analysis in State Space For the case where the system has multiple inputs and multiple outputs, ...

Section 9–3 / Transformation of System Models with MATLAB 659 EXAMPLE 9–4 Consider a system with multiple inputs and multiple ou ...

660 Chapter 9 / Control Systems Analysis in State Space 9–4 Solving the Time-Invariant State Equation In this section, we shall ...

Section 9–4 / Solving the Time-Invariant State Equation 661 By substituting this assumed solution into Equation (9–28), we obtai ...

662 Chapter 9 / Control Systems Analysis in State Space Because of the convergence of the infinite series the series can be diff ...

Section 9–4 / Solving the Time-Invariant State Equation 663 Hence, The difference between and vanishes if AandBcommute. Laplace ...

664 Chapter 9 / Control Systems Analysis in State Space (The inverse Laplace transform of a matrix is the matrix consisting of t ...

Section 9–4 / Solving the Time-Invariant State Equation 665 If there is a multiplicity in the eigenvalues—for example, if the ei ...

666 Chapter 9 / Control Systems Analysis in State Space Hence, Noting that we obtain the inverse of the state-transition matrix ...

Section 9–4 / Solving the Time-Invariant State Equation 667 Integrating the preceding equation between 0 and tgives or (9–41) Eq ...

668 Chapter 9 / Control Systems Analysis in State Space EXAMPLE 9–6 Obtain the time response of the following system: whereu(t)i ...

Section 9–5 / Some Useful Results in Vector-Matrix Analysis 669 where Since (lI-A)adj(lI-A)=Cadj(lI-A)D(lI-A)=|lI-A|I we obtain ...