Lecture Notes on Linear System Theory

Lecture Notes on Linear System Theory John Lygeros∗ and Federico A. Ramponi† ∗Automatic Control Laboratory, ETH Zurich CH-8092, Zurich, Switzerland lygeros@control.ee.ethz.ch †Department of Information Engineering, University of Brescia Via Branze 38, 25123, Brescia, Italy federico.ramponi@unibs.it January 3, 2015 Contents 1 Introduction 1 1.1 Objectives of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Proof methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Functions and maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Introduction to Algebra 11 2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Rings and fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Subspaces and bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Linear maps generated by matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Matrix representation of linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.8 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Introduction to Analysis 33 3.1 Norms and continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Equivalent norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Infinite-dimensional normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Induced norms and matrix norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7.1 Background lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7.2 Proof of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.7.3 Proof of uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Time varying linear systems: Solutions 59 4.1 Motivation: Linearization about a trajectory . . . . . . . . . . . . . . . . . . . . . . 59 i 4.2 Existence and structure of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 State transition matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5 Time invariant linear systems: Solutions and transfer functions 70 5.1 Time domain solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Semi-simple matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3 Jordan form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.4 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Stability 85 6.1 Nonlinear systems: Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2 Linear time varying systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.3 Linear time invariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4 Systems with inputs and outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5 Lyapunov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7 Inner product spaces 104 7.1 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.2 The space of square-integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 Orthogonal complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.4 Adjoint of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.5 Finite rank lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.6 Application: Matrix pseudo-inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8 Controllability and observability 118 8.1 Nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.2 Linear time varying systems: Controllability . . . . . . . . . . . . . . . . . . . . . . . 121 8.3 Linear time varying systems: Minimum energy control . . . . . . . . . . . . . . . . . 124 8.4 Linear time varying systems: Observability and duality . . . . . . . . . . . . . . . . 127 8.5 Linear time invariant systems: Observability . . . . . . . . . . . . . . . . . . . . . . . 131 8.6 Linear time invariant systems: Controllability . . . . . . . . . . . . . . . . . . . . . . 135 8.7 Kalman decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9 State Feedback and Observer Design 139 9.1 Revision: Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.2 Linear state feedback for single input systems . . . . . . . . . . . . . . . . . . . . . . 141 9.3 Linear state observers for single output systems . . . . . . . . . . . . . . . . . . . . . 146 9.4 Output feedback and the separation principle . . . . . . . . . . . . . . . . . . . . . . 149 9.5 The multi-input, multi-output case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 ii A Notation 158 A.1 Shorthands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B Basic linear algebra 160 C Basic calculus 161 iii Chapter 1 Introduction 1.1 Objectives of the course This course has two main objectives. The first (and more obvious) is for students to learn something about linear systems. Most of the course will be devoted to linear time varying systems that evolve in continuous time t ∈ R+. These are dynamical systems whose evolution is defined through state space equations of the form x˙(t) = A(t)x(t) +B(t)u(t), y(t) = C(t)x(t) +D(t)u(t), where x(t) ∈ Rn denotes the system state, u(t) ∈ Rm denotes the system inputs, y(t) ∈ Rp denotes the system outputs, A(t) ∈ Rn×n, B(t) ∈ Rn×m, C(t) ∈ Rp×n, and D(t) ∈ Rp×m are matrices of appropriate dimensions, and where, as usual, x˙(t) = dx(t) denotes the derivative of x(t) with respect to time. Time varying linear systems are useful in many application areas. They frequently arise as models of mechanical or electrical systems whose parameters (for example, the stiffness of a spring or the inductance of a coil) change in time. As we will see, time varying linear systems also arise when one linearizes a non-linear system around a trajectory. This is very common in practice. Faced with a nonlinear system one often uses the full nonlinear dynamics to design an optimal trajectory to guide the system from its initial state to a desired final state. However, ensuring that the system will actually track this trajectory in the presence of disturbances is not an easy task. One solution is to linearize the nonlinear system (i.e. approximate it by a linear system) around the optimal trajectory; the approximation is accurate as long as the nonlinear system does not drift too far away from the optimal trajectory. The result of the linearization is a time varying linear system, which can be controlled using the methods developed in this course. If the control design is done well, the state of the nonlinear system will always stay close to the optimal trajectory, hence ensuring that the linear approximation remains valid. A special class of linear time varying systems are linear time invariant systems, usually referred to by the acronym LTI. LTI systems are described by state equations of the form x˙(t) = Ax(t) +Bu(t), y(t) = Cx(t) +Du(t), where the matrices A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, and D ∈ Rp×m are constant for all times t ∈ R+. LTI systems are somewhat easier to deal with and will be treated in the course as a special case of the more general linear time varying systems. 1 ... - tailieumienphi.vn