INTRODUCTION TO LINEAR SYSTEM THEORY

Keypoints:

• INTRODUCTION TO LINEAR SYSTEM THEORY
  • Linear Dynamical System Representation
  • Linearization and LTI State Equation Solutions
  • LTI Response Properties: Transfer Function Poles and Zeros
  • LTI Response Properties: Steady-State Frequency Response
  • LTI Response Properties: Matrix Forms

This file is based on ME 530.616 Introduction to Linear Systems Theory by Prof. Louis Whitcomb

Linear Dynamical System Representation

• Linear Function: The function $f$ is linear if and only if $f(x+y)=f(x)+f(y)$ and $f(ax)=af(x)$.

• Linear Dynamical System Representations:
  • Convolution Integral
  • Laplace Transform
  • n-th Order Differential Equation
  • n-dimensional State Space Representation
    • LTV: Linear Time-Varying $\dot x(t)=A(t)x(t)+B(t)u(t)$ $y(t)=C(t)x(t)+D(t)u(t)$
    • LTI: Linear Time-Invariant $\dot x(t)=Ax(t)+Bu(t)$ $y(t)=Cx(t)+Du(t)$, where $A,B,C,D$ are constant matrices.
• Important questions about linear dynamical systems:
  • Boundeness: Do all signals remain bounded?
  • Stability: Do all signals converge to zero?
  • Controllability: Can we drive the system to any state?
  • Observability: Can we determine the state from the output?
  • Realization: Given an input-output map, can we find a state-space representation?

Linearization and LTI State Equation Solutions

• Linear approximation of nonlinear dynamical system at an equilibrium point: the second-order Taylor series expansion of the nonlinear function $f$ about the equilibrium point $(x_0,u_0)$.
  • $\dot x=f(x,u)$ $f(x,u)=f(x_0,u_0)+\frac{\partial f}{\partial x}\vert_{x=x_0,u=u_0}(x-x_0)+\frac{\partial f}{\partial u}\vert_{x=x_0,u=u_0}(u-u_0)+\mathcal{O}(|x-x_0|^2+|u-u_0|^2)$ $\dot x=A(x-x_0)+B(u-u_0)+\mathcal{O}$, where $A=\frac{\partial f}{\partial x}\vert_{x=x_0,u=u_0}$ and $B=\frac{\partial f}{\partial u}\vert_{x=x_0,u=u_0}$.
  • Stability of linearized system: the real parts $\lambda_i$ of the eigenvalues of $A$ determine the stability of the linearized system.
    • All negative: asymptotically stable
    • Some zero: indeterminate
      • $Re(\lambda_i)=0$, and algebraic multiplicity=geometric multiplicity: marginally stable
    • Any positive: unstable
• LTI State Equation Solutions:
  • Zero-input case: $u(t)=0$
    • $\dot x=Ax$ $x(t)=e^{At}x(0)=\sum_{i=0}^{\infty}\frac{A^it^i}{i!}x(0)$
    • $e^{At}=\mathcal{L}^{-1}[(sI-A)^{-1}]$
  • Nonzero-input case: $u(t)\neq 0$
    • $\dot x=Ax+Bu$ $x(t)=e^{At}x(0)+\int_0^te^{A(t-\tau)}Bu(\tau)d\tau$

LTI Response Properties: Transfer Function Poles and Zeros

$u(s) \rightarrow G(s) \rightarrow y(s)$, where $G(s)$ is the transfer function i.e $G(s)=\frac{Y(s)}{U(s)}$.

• Time domain state space representation: $\dot x=Ax+Bu$, $y=Cx+Du$
• Frequency domain transfer function representation:
  • $G(s)=C(sI-A)^{-1}B+D$
  • $G(s)=\frac{n(s)}{d(s)} + D = \frac{b_ms^m+b_{m-1}s^{m-1}+…+b_1s+b_0}{s^n+a_{n-1}s^{n-1}+…+a_1s+a_0} + D$
  • Poles are the roots of the denominator polynomial $d(s)$, and zeros are the roots of the numerator polynomial $n(s)$.
  • Poles are the eigenvalues of $A$, but eigenvalues of $A$ are not necessarily poles of $G(s)$ due to pole-zero cancellation. Because $(sI-A)^{-1} = \frac{adj(sI-A)}{det(sI-A)}$, and $det(sI-A)$ is the characteristic polynomial of $A$.

LTI Response Properties: Steady-State Frequency Response

• Steady-State Response: given a LTI system with response $y(t)$ to initial condition $x(t_0)=x_0$ and input $u(t)$, the steady-state response $y_{ss}(t)$ is that function satisfying $\lim_{t\rightarrow\infty}[y(t)-y_{ss}(t)]=0$.
  • If $y_{ss}(t)$ is constant, it’s regarded as the final value.
  • $y_{ss}(t)$ may not be constant.
  • $y_{ss}(t)$ may oscillate.
  • $y_{ss}(t)$ may be unbouded.
  • Final value theorem: $\lim_{t\rightarrow\infty}y(t)=\lim_{s\rightarrow 0}sY(s)$, if $\lim_{t\rightarrow\infty}y(t)$ exists and is finite.

LTI Response Properties: Matrix Forms

• A square matrix $A$ is diagonalizable if there exists an invertible matrix $T$ such that $T^{-1}AT=\Lambda$, where $\Lambda$ is a diagonal matrix.

• A square matrix $A$ is diagonalizable if and only if it has $n$ linearly independent eigenvectors.
• A matrix is diagonalizable if and only if all its eigenvalues $\lambda_i$, their algebraic multiplicities $m_i$, and their geometric multiplicities $g_i$ satisfy $m_i=g_i$.

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