Steady-State Equilibria Modeling from first principles Linearisation
What do we want in control?
Stability: With no disturbances and constant input, output should approach a desired, constant equilibrium, even if initial conditions change slightly from nominal values
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Transient performance: quick but “graceful” transition from initial value to steady-state.
Robustness: Steady-state behaviour doesn’t change dramatically with uncertainties, disturbances or noise.
What is an equilibrium?
Equilibria
When a system is exactly at an equilibrium point, nothing changes over time.
Can either be stable, marginally stable, or unstable
Why Go for Equilibria?
Equilibria are “sweet spots”. When system input and output variables start from an equilibrium point, they remain there forever, if no disturbances and noises.
Nominal operating conditions should be chosen to be equilibrium points.
In terms of the whole control system, we want to operate near equilibrium points that are stable.
On Unstable Equilibria
However when we consider just the plant, we sometimes operate at plant equilibria that are unstable.
We render these points stable by interconnecting the plant with a feedback controller.
Preview of Techniques
Stability at equilibrium studied using Laplace transforms, i.e. s-domain, after linearisation
Transient performance examined in s-domain or frequency domain (Fourier transforms)
Performance and robustness will be investigated in freq. domain. System freq. response typically needs
Good tracking of low freq. references
Good damping of disturbances and high freq. noise
System models
Continuous-time signal
Continuous-time signal
Laplace transform
Fourier transform
Algebraic/ Differential eq./ Convolution with impulse resp. h(.)
Laplace transform
Transfer function
Fourier transform
Frequency response H(jω)
System = Mapping between Signal Spaces
How to construct a model
From first principles (i.e. laws of nature), often followed by
– Linearisation
– Model reduction (e.g. nulling small coefficients)
Collecting lots of empirical data about input- output signal values over time and then fitting a convenient model (system identification, machine learning)
All models are wrong, but some are useful.
– G. Box (statistician)
A model should be as simple as possible, but no simpler. – A. Einstein (…)
Simple models of the plant are often more useful than complex ones, even if they’re less accurate:
– Each model parameter easier to estimate and interpret from empirical data
– Well-designed feedback law attenuates the effect of the modelling error
Static models
Input u(t)
Output y(t)
Algebraic eq.
Property: If u(t)= u(t’) then y(t) = y(t’)
Example: Ohm’s law for resistor
Assumptions:
Resistor is made of “Ohmic” material Temperature is constant
The current is sufficiently small
Ageing is ignored, etc.
Example: Hooke’s law for spring
Assumptions:
The deflections are sufficiently small so that the material stays within its linear elasticity region (linearization!).
Differential equation models
Continuous time signal
Continuous time signal
Differential eq.
Property : y(t) not necessarily = y(t’) when u(t) =u(t’). Example:
Example (mass with spring)
Assumptions:
Hooke’s law
Newton’s second law No friction
No damping
Example (electrical):
Assumptions:
Ohm’s law
Ideal capacitor
Ideal inductor
Kirchoff’s current and voltage laws
Example (fluid flow):
Assumptions:
Law of conservation of mass
Constant cross section A
Atmospheric pressure above water and at the outlet, and so on.
Note the time delay and nonlinearity!
Example (pendulum):
Assumptions:
Bob is a point mass
Arm is rigid but massless No friction, etc
Example (pendulum)
Free body diagram:
Newton’s second law (horizontal): Newton’s second law (vertical)
Example (pendulum)
Eliminating the force in the arm yields: Note that this model is nonlinear!
We will revisit this model when we talk about equilibria and linearisation.
Example (DC motor):
(Linearized) Motor: Newton’s second law: Kirchoff’s voltage law:
Finding Equilibria
Consider an input-output differential model
A signal pair Iis an equilibrium if
Equilibria can be computed by solving the static equation
Example (pendulum)
Consider the pendulum equation
where we think of as the output. Equilibria satisfy:
Example (pendulum)
If we are interested only in equilibria with then, we have
When mass is down: When mass is up:
Why Think Linear?
Often your plant model is already linear time- invariant (LTI).
But sometimes, you’ll get a nonlinear model Difficult to fully analyse and control
In the philosophy of starting simple, your first step should be to linearise the plant model around an equilibrium of interest.
If disturbances are not too big, this will often yield satisfactory linear controller designs
Linearising a function
NOTE: linearisation of the same function around different points is different!
Linearizing a nonlinear function means approximating it around a chosen operating point by its tangent line
Yields a good approximation valid near this point
Linearisation is different around different points, because the tangent lines will be different
Linearizing a nonlinear function means approximating it around a chosen operating point by its tangent line
Yields a good approximation valid near this point
Linearisation is different around different points, because the tangent lines will be different
Taylor series expansion
To linearise a function near a point a, find its Taylor series
then ignore the 2nd and higher order terms
Yields a good approximation for small deviation
Linearization
Linearization of this function around x=a is:
Graphical representation
Linearising a scalar ( and static) input-output system
Suppose output and input signals satisfy l(y(t),u(t)) = 0.
We first select an equilibrium of interest
Then find Taylor series around this equilibrium, keeping only 1st order terms
Linearised Scalar Static System
Linearised model is
Note that linearized model is different for
different equilibria.
Note that the following model is affine, not strictly linear:
Note: we assume that the derivative w.r.t. y does not vanish at equilibrium.
We consider the multivariate function
(each subscript denotes time-derivative of that order in the
relevant signal)
We find its 1st order Taylor series around
This is a good approximation for small Linearized ODE is a map:
that is obtained by noting
Example (pendulum)
From the differential equation, we see We linearise the function around an eq.
Example (pendulum)
We define incremental input and output and obtain a linear incremental ODE model:
Exercise: what are linearisations around and
Equilibria can be obtained by solving an algebraic equation.
A system can have multiple equilibria.
The system can be approximated by a linear
system around an equilibrium.
Linearisation is different around different equilibria.
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