April 9th 2022
Linear Dynamical Systems
Freyr Hlynsson - freyr19@ru.is
Hrafn Sölvi Sigurðarson - hrafn19@ru.is
Linear dynamical system
● Dynamical system: A collection of variables that describes the state of an
object or collection of objects that vary with time
● Linear: variables are only multiplied by constants and added together
u
a
b
Representing the system as a state space
representation
Transforming to state space
Only care for the temperature in room 2
Equilibrium Points
● A system is at an equilibrium point when it does not change with time
● Find an x such that:
For example for our system:
And solve for all T
Stability of a system
● The eigenvalues of the A matrix in state space determines the stability
● Eigenvalues are found by solving the following equation for λ
● The system is asymptotically stable if the real parts of all eigenvalues are
negative
Determinant of a matrix
● 2x2 Matrix
● 3x3 matrix
Determinant of 3x3 matrix
Stability
● One can either
○ calculate all the eigenvalues and check if they are negative
○ Use the Routh-Hurwitz criterion
Routh-Hurwitz
● With the (characteristic) polynomial obtained from
● Polynomial:
Condition:
Analytical Solution
● The analytical solution is given by
● v - eigenvector with corresponding eigenvalue λ, x0 is the initial value
Numerical solution (integration)
● The Euler method is valid if the time step is sufficiently small, and dictates that
● System behaviour can be estimated without needing an analytical solution
Nonlinear dynamics
● Calculate the Jacobian to linearise
● Each differential equation is partially differentiated with respect to each
variable, and the result put in the corresponding entry of A
Control
● Many values one can assign u
○ A constant
○ A function
○ …
● PID - controller
system error
PID - controller
● Proportional: immediate response to error
● Integral: removes steady state error
● Derivative: dampens system, as well as speeds up response
PID in state space example
Lets say, a = 2
and b = 3
Stability of PID controller
To find the stability we calculate the determinant using our new A:
We can use the Routh-Hurwitz Criteria to determine the gain values to create a stable
system
Performance Measures
● Stability is a very basic measure of performance
● Overshoot, undershoot, risetime, settling time
# Graf
Transfer functions
● A more powerful way of analysing a system is though the transfer function
● The transfer function can be derived by a couple of methods
○ From state space
○ From a differential equations
○ From block diagrams
Transfer function from state space
● Obtained by computing
● Generally takes the form
● b(s) is the same characteristic polynomial obtained through eigenvalue
calculation
Transfer function from state space our
example
Lets say, a = 2 and b = 3
=>
Using PID in frequency domain
Stability and Pole placement
To have the system stable we focus on the poles meaning what s is when:
Lets say we want the poles to be at s=-4, s= -5, s=-6 then we can find the following:
And now match our equation to the equation created:
From differential equations
● The matrices A, B, and C can be read from the differential equation
● Here A= -a, B=1 and C=1
From a block diagram
● A block diagram is a visual way of representing a control system structure
● Each transfer function is given, and the connections between them determine
the final transfer function
PID Tuning - initial guess
Windup
● Actuators (things that influence a system) have limits
○ Maximum horsepower, turn, thrust, etc
● PID controller will think it is getting more input than is actually happening
● Will never reach steady state
Anti-Windup
● Stop the I part from integrating the error when saturation occurs
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