# Linear Systems

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Previous chapters have focused on the dynamics of a system with relatively little attention to the inputs and outputs. This chapter gives an introduction to input/output behavior for linear systems and shows how a nonlinear system can be approximated near an equilibrium point by a linear model.

## Chapter Summary

1. A linear system is one in which the output is jointly linear in the intitial condition for the system and the input to the system. In particular, a linear system has the property that if we apply an input ${\displaystyle u(t)=\alpha u_{1}(t)+\beta u_{2}(t)}$ with zero initial condition, the corresponding output will be ${\displaystyle y(t)=\alpha y_{1}(t)+\beta y_{2}(t)}$, where ${\displaystyle y_{i}}$ is the output associated with the input ${\displaystyle u_{i}}$. This propery is called linear superposition.

2. A differential equation of the form

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu&\quad x&\in R^{n},u\in R\\y&=Cx+Du&y&\in R\end{aligned}}}

is a single-input, single-output (SISO) linear differential equation. Its solution can be written in terms of the matrix exponential

${\displaystyle e^{At}=I+At+{\frac {1}{2}}A^{2}t^{2}+{\frac {1}{3!}}A^{3}t^{3}+\cdots =\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k}t^{k}.}$

The solution to the differential equation is given by the convolution equation

${\displaystyle y(t)=Ce^{At}x(0)+\int _{0}^{t}Ce^{A(t-\tau )}Bu(\tau )d\tau +Du(t).}$

3. A linear system

${\displaystyle {\dot {x}}=Ax}$
is asymptotically stable if and only if all eigenvalues of ${\displaystyle A}$ all have strictly negative real part and is unstable if any eigenvalue of ${\displaystyle A}$ has strictly positive real part. For systems with eigenvalues having zero real-part, stability is determined by using the Jordan normal form associated with the matrix. A system with eigenvalues that have no strictly positive real part is stable if and only if the Jordan block corresponding to each eigenvalue with zero part is a scalar (1x1) block.

4. The input/output response of a (stable) linear system contains a transient region portion, which eventually decays to zero, and a steady state portion, which persists over time. Two special responses are the step response, which is the output corresponding to an step input applied at ${\displaystyle t=0}$ and the frequency response, which is the response of the system to a sinusoidal input at a given frequency.

5. The step response is characterized by the following parameters:

• The steady state value, ${\displaystyle y_{\text{ss}}}$, of a step response is the final level of the output, assuming it converges.
• The rise time, ${\displaystyle T_{\text{r}}}$, is the amount of time required for the signal to go from 10% of its final value to 90% of its final value.
• The overshoot, ${\displaystyle M_{\text{p}}}$, is the percentage of the initial value by which the signal initially rises above the final value.
• The settling time, ${\displaystyle T_{\text{s}}}$, is the amount of time required for the signal to stay within 5% of its final value for all future times.

6. The frequency response is given by

${\displaystyle y(t)=\underbrace {Ce^{At}{\Bigl (}x(0)-(sI-A)^{-1}B{\Bigr )}} _{\text{transient}}+\underbrace {{\Bigl (}D+C(sI-A)^{-1}B{\Bigr )}e^{st}} _{\text{steady state}},}$

where ${\displaystyle \cos \omega t={\frac {1}{2}}\left(e^{j\omega t}+e^{-j\omega t}\right)}$ and ${\displaystyle s=i\omega }$. The gain and phase of the frequency response are given by

${\displaystyle {\text{gain}}(\omega )={\frac {A_{y}}{A_{u}}}=M\qquad {\text{phase}}(\omega )=\phi -\psi =\theta .}$

7. A nonlinear system of the form

{\displaystyle {\begin{aligned}{\dot {x}}&=f(x,u)&\quad x&\in \mathbb {R} ^{n},u\in \mathbb {R} \\y&=h(x,u)&y&\in \mathbb {R} \end{aligned}}}

is a single-input, single-output (SISO) nonlinear system. It can be linearized about an equibrium point ${\displaystyle x=x_{\text{e}}}$, ${\displaystyle u=u_{\text{e}}}$, ${\displaystyle y=y_{\text{e}}}$ by defining new variables

${\displaystyle z=x-x_{\text{e}}e\qquad v=u-u_{\text{e}}\qquad w=y-h(x_{\text{e}},u_{\text{e}}).}$

The dynamics of the system near the equilibrium point can then be approximated by the linear system

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu,\\y&=Cx+Du,\end{aligned}}}

where

{\displaystyle {\begin{aligned}A&=\left.{\frac {\partial f(x,u)}{\partial x}}\right|_{x_{\text{e}},u_{\text{e}}}&\quad B&=\left.{\frac {\partial f(x,u)}{\partial u}}\right|_{x_{\text{e}},u_{\text{e}}}\\C&=\left.{\frac {\partial h(x,u)}{\partial x}}\right|_{x_{\text{e}},u_{\text{e}}}&\quad D&=\left.{\frac {\partial y(x,u)}{\partial u}}\right|_{x_{\text{e}},u_{\text{e}}}\end{aligned}}}

The equilibrium point for a nonlinear system is locally asymptotically stable if the real part of the eigenvalues of the linearization about that equilibrium point have strictly negative real part.

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### Python Code

The following Python scripts are available for producing figures that appear in this chapter.

See the software page for more information on how to run these scripts.