ABSTRACT
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.5 Statement of the Regulation Problem . . . . . . . . . . . . . . . . . . . 9
1.2 Main Theoretical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Solvability Criteria for Regulator Equations . . . . . . . . . . . . 14 1.4 SISO Examples with Bounded Control and Sensing . . . . . . . . . . . . 16
1.4.1 Set-Point Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.2 Set-Point Disturbance Rejection . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.3 Harmonic Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.4 Harmonic Disturbance Rejection . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 The MIMO Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.1 Problem Setting and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.2 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5.3 Solution of the MIMO Regulator Equations Using the
Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Systems
In the first part of this book we focus on regulation problems for linear distributed parameters systems with bounded input and output operators. As we have already mentioned in the introduction our main interest is in applications to systems governed by partial differential equations. For such systems the case of bounded input and output operators does not contain the most interesting applications. However, many unbounded operators have bounded approximations that are widely considered in the literature and since the historical development begins with this case that’s where we begin. Moreover, this case most closely parallels the classical results for finite-dimensional linear systems. Although we are interested in applications to control systems governed by PDEs, we choose to present our basic model as an abstract control system
zt = Az +Bdd+Binu, (1.1)
z(0) = z0, z0 ∈ Z, (1.2) y(t) = Cz(t), (1.3)
where z ∈ Z is the state of the system evolving in Z, a complex separable Hilbert space (state space) with inner product denoted by 〈·, ·〉 and norm ‖ϕ‖ = 〈ϕ,ϕ〉1/2. We use z0 to denote the initial condition, y(t) the measured output, and u(t) is the control input. Here u ∈ U the input space and y ∈ Y the output space where U and Y are finite-or infinite-dimensional Hilbert input and output spaces, respectively. A is an unbounded densely defined operator, with domain D(A) in Z, and it is assumed to be the infinitesimal generator of a strongly continuous semigroup (see, for example, [34, 72]) on Z, Bd ∈ L(U,Z) and C ∈ L(Z,Y). The symbol L(W1,W2) denotes the set of all bounded linear operators from a Hilbert space W1 to a Hilbert space W2. When W = W1 = W2 we write L(W ). The term d(t) represents a disturbance. In general Bd refers to a disturbance input operator and Bin refers to a control input operator.
