Instructions
ELEN90055 Control Systems Worksheet 1
Semester 2
This worksheet covers materials about Equilibria and Linearisation from lectures 5 and 6.
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Review of key concepts
Consider a system described by its model:
dny dy dnu du l dtn ,…, dt,y, dtn ,…, dt,u
Equilibria are calculated by considering: l(0,…,0,y ̄,0,…,0,u ̄) = 0 .
To perform linearisation, we find it convenient to consider the multivariate function: l(yn,…,y0,un …,u0) .
We obtain the linearisation of the above equation around the chosen equilibrium (0, . . . , 0, y ̄, 0, . . . , 0, u ̄): llin (yn,…y1,y0 −y ̄,un …,u1,u0 −u ̄)
and diδu dti
∂yn (0,…,0,y ̄,0,…,0,u ̄)
·(yn −0)+…+ ∂l ·(y1 −0) ∂y1 (0,…,0,y ̄,0,…,0,u ̄)
∂y0 (0,…,0,y ̄,0,…,0,u ̄)
∂u1 (0,…,0,y ̄,0,…,0,u ̄)
·(y0−y ̄)+ ∂l ·(un−0)+… ∂un (0,…,0,y ̄,0,…,0,u ̄)
·(u1 −0)+ ∂l ·(u0 −u ̄). ∂u0 (0,…,0,y ̄,0,…,0,u ̄)
(3) = di y
Introducing the incremental variables δ := y − y ̄, δ := u − u ̄, and noting that di δy
= diu, the linearised model is given by: dti
dnδy dδy dnδu dδu llin dtn ,…, dt ,δy, dtn ,…, dt ,δu
Tutorial problems
Consider the water tank from Lecture 5. The model of the change of water level in the tank can be written as:
dV = d(Ax) =Adx =qi −qo , (4) dt dt dt
where x is the level of water in the tank, A is the area of the cross-section of the tank (assumed constant) and qi and qo are the volumetric flows into and out of the tank respectively. qi is assumed to be the input to the system and x the output. Moreover, suppose that the outflow is given by the following relation:
qo=k x. (5) Find all equilibria and linearise the system around the equilibrium where x = 1 [m];
assume that A = 1 [m2] and k = 2 [m5/2/s].
Consider a mass m sliding on a horizontal surface and attached to a vertical surface through a spring, see Figure 1. The mass is subjected to an external force F (the input to the system), the spring force and a friction force Ff . We define y as the displacement from a reference position (the output of the system). We assume that Ff = cy ̇ and we will consider two models for the spring force:
• softening spring model Fsp = k(1 − a2y2)y, |ay| ≤ 1; • hardening spring model Fsp = k(1 + a2y2)y,
which are typically valid for large displacements y and for different types of springs. Newton’s law of motion is given by:
my ̈+Ff +Fsp =F . Assume that m = 1kg,c = 1kg/s,k = 1N/m,a = 10.
• Find all equilibria for the system with softening spring. Linearise the system around equilibrium for which y = 0.01m;
• Find all equilibria for the system with hardening spring. Linearise the system around equilibrium for which y = −0.01m.
Consider the Lotka-Volterra or predator-prey equations, which model the change in predator and prey populations over time in a closed ecology:
x ̇ = αx−βxy y ̇ = δxy−γy,
Figure 1: A mass-spring system.
where α, β, γ, δ are constants that depend on the particular predator-prey system, x denotes the number of prey (e.g. rabbits) and y denotes the number of predators (e.g. foxes). Find all equilibria of this system and discuss the solution. Suppose that α = 2/3, β = 4/2, γ = δ = 1. Linearise the system around each equilibrium.
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