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 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.
2. 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
The solution to the differential equation is given by the convolution equation
3. A linear systemis 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 blockcorresponding 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 thestep 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.
5. 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.
6. The frequency response is given by
where and s = jω. The gain and phase of the frequency response are given by
7. A nonlinearsystem 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
The dynamics of the system near the equilibrium point can then be approximated by the linear system
The equilibrium point for a nonlinear system is locally asymptotically stable if the real part of theeigenvalues of the linearization about that equilibrium point have strictly negative real part.
this chapter describes how state feedback can be used to design the (closed loop) dynamics of the system:
1. A linear system with dynamics
is said to be reachable if we can find an input u(t) defined on the interval [0,T] that can steer the system from a given final point x(0) = x0 to a desiredfinal point x(T) = xf.
2. The reachability matrix for a linear system is given by
A linear system is reachable if and only if the reachability matrix Wr is invertible (assuming a single intput/single output system). Systems that are not reachable have states that are constrained to have a fixed relationship with each other.
3. A linear system of the form
is said to be in reachablecanonical form. A system in this form is always reachable and has a characteristic polynomial given by
A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation z = Tx.
4. A state feedback law has the form
where r is the reference value for the output. The closed loop dynamics for the system are given by
The stability of the...