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.
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Chapter Summary
This chapter introduces the analysis tools for linear input/output systems.
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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 u(t) = αu1(t) + βu2(t) with zero initial condition, the corresponding output will be y(t) = αy1(t) + βy2(t), where yi is the output associated with the input ui. This propery is called linear superposition.
A differential equation of the form
is a single-input, single-output (SISO) linear differential equation. Its solution can be written in terms of the matrix exponential
A linear system
is asymptotically stable if and only if all eigenvalues of A all have strictly negative real part and is unstable if any eigenvalue of 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 strickly positive real part is stable if and only if the Jordan block corresponding to each eigenvalue with zero part is a scalar (1x1) block.
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 t = 0 and the frequency response, which is the response of the system to a sinusoidal input at a given frequency.
The step response is characterized by the following parameters:
- The steady state value, yss, of a step response is the final level of the output, assuming it converges.
- The rise time, Tr, 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, Mp, is the percentage of the infal value by which the signal initially rises above the final value.
- The settling time, Ts, is the amount of time required for the signal to stay within 5% of its final value for all future times.
The frequency response is given by
where
and s = jω. The gain and phase of the frequency response are given by
A nonlinear system of the form
is a single-input, single-output (SISO) nonlinear system. It can be linearized about an equibrium point x = xe, u = ue, y = ye by defining new variables
.
where
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.
Exercises
Frequently Asked Questions
- FAQ: Are the percentages in the definition of rise time, overshoot measured from the final value, or the size of the input change?
- FAQ: How do you show that exp(T S inv(T)) = T exp(S) inv(T)?
- FAQ: In Example 5.3, why is the A matrix in the given form?
- FAQ: What is an example of a system with Re(λ)=0 that is not stable? What if Im(λ) is not zero?
Errata
- Errata: Example 5.5 includes a damping term not shown in Figure 5.4
- Errata: Extraneous text "!linear" in Section 5.1
- Errata: In Examples 5.5 and 5.6, dynamics in modal form should use z, not x
- Errata: In Exercise 5.7, the 5% settling time should be 3 tau instead of 2 tau
- Errata: In equation (5.5), the upper limit of the integral should be t
- Errata: Parameter values are missing in Exercise 5.3
