Chapter 5 State and Output Feedback This chapter describes how feedback can be used shape the local behavior of a system Both state and output feedback are discussed The concepts of reachability and observability are introduced and it is shown how states can be estimated from measurements of the input and the output 5 1 Introduction The idea of using feedback to shape the dynamic behavior was discussed in broad terms in Section 1 4 In this chapter we will discuss this in detail for linear systems In particular it will be shown that under certain conditions it is possible to assign the system eigenvalues to arbitrary values by feedback allowing us to design the dynamics of the system The state of a dynamical system is a collection of variables that permits prediction of the future development of a system In this chapter we will explore the idea of controlling a system through feedback of the state We will assume that the system to be controlled is described by a linear state model and has a single input for simplicity The feedback control will be developed step by step using one single idea the positioning of closed loop eigenvalues in desired locations It turns out that the controller has a very interesting structure that applies to many design methods This chapter may therefore be viewed as a prototype of many analytical design methods If the state of a system is not available for direct measurement it is often possible to determine the state by reasoning about the state through our knowledge of the dynamics and more limited measurements This is done by building an observer that uses measurements of the inputs and outputs of a linear system along with a model of the system dynamics to 109 110 CHAPTER 5 STATE AND OUTPUT FEEDBACK estimate the state The details of the analysis and designs in this chapter are carried out for systems with one input and one output but it turns out that the structure of the controller and the forms of the equations are exactly the same for systems with many inputs and many outputs There are also many other design techniques that give controllers with the same structure A characteristic feature of a controller with state feedback and an observer is that the complexity of the controller is given by the complexity of the system to be controlled Thus the controller actually contains a model of the system This is an example of the internal model principle which says that a controller should have an internal model of the controlled system 5 2 Reachability We begin by disregarding the output measurements and focus on the evolution of the state which is given by dx Ax Bu dt 5 1 where x Rn u R A is an n n matrix and B an n 1 matrix A fundamental question is if it is possible to find control signals so that any point in the state space can be reached First observe that possible equilibria for constant controls are given by Ax bu0 0 This means that possible equilibria lies in a one or possibly higher dimensional subspace If the matrix A is invertible this subspace is spanned by A 1 B Even if possible equilibria lie in a one dimensional subspace it may still be possible to reach all points in the state space transiently To explore this we will first give a heuristic argument based on formal calculations with impulse functions When the initial state is zero the response of the state to a unit step in the input is given by x t Z t eA t Bd 5 2 0 The derivative of a unit step function is the impulse function t which may be regarded as a function which is zero everywhere except at the origin 5 2 REACHABILITY 111 and with the property that Z t dt 1 The response of the system to a impulse function is thus the derivative of 5 2 dx eAt B dt Similarly we find that the response to the derivative of a impulse function is d2 x AeAt B dt2 The input u t 1 t 2 t t n n 1 t thus gives the state x t 1 eAt B 2 AeAt B 3 A2 eAt B n An 1 eAt B Hence right after the initial time t 0 denoted t 0 we have x 0 1 B 2 AB 3 A2 B n An 1 B The right hand is a linear combination of the columns of the matrix Wr B AB An 1 B 5 3 To reach an arbitrary point in the state space we thus require that there are n linear independent columns of the matrix Wc The matrix is called the reachability matrix An input consisting of a sum of impulse functions and their derivatives is a very violent signal To see that an arbitrary point can be reached with smoother signals we can also argue as follow Assuming that the initial condition is zero the state of a linear system is given by Z t Z t A t eA Bu t d e Bu d x t 0 0 It follows from the theory of matrix functions that eA I 0 A 1 An 1 n 1 112 CHAPTER 5 STATE AND OUTPUT FEEDBACK and we find that Z t Z t 1 u t d 0 u t d AB x t B 0 0 Z t n 1 u t d An 1 B 0 Again we observe that the right hand side is a linear combination of the columns of the reachability matrix Wr given by 5 3 We illustrate by two examples Example 5 1 Reachability of the Inverted Pendulum Consider the inverted pendulum example introduced in Example 3 5 The nonlinear equations of motion are given in equation 3 5 dx x2 sin x1 u cos x1 dt y x1 Linearizing this system about x 0 the linearized model becomes dx 0 0 1 u x 1 1 0 dt y 1 0 x 5 4 The dynamics matrix and the control matrix are 0 1 0 A B 1 0 1 The reachability matrix is 0 1 Wr 1 0 5 5 This matrix has full rank and we can conclude that the system is reachable This implies that we can move the system from any initial state to any final state and in particular that we can always find an input to bring the system from an initial state to the equilibrium Example 5 2 System in Reachable Canonical Form Next we will consider a system by in reachable canonical form a1 a2 an 1 an 1 1 0 0 0 0 dz 1 0 0 0 z 0 u A z B u dt 0 0 1 0 0 5 2 REACHABILITY 113 To show that Wr is full rank we show that the inverse of the reachability matrix exists and is given by 1 a 1 a2 an 0 1 a1 an 1 1 5 6 W r 0 0 0 1 To show this we consider …
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