State Space Control

• When should feedback control be applied? If and only if there is uncertainty (unknown disturbances, modelling errors)
• What is the purpose of feedback control? To attenuate the effects of uncertainty on desired system outputs
• Some of he drawbacks of feedback control:

  Can destabilise a system

  Couples different parts of the system (may inject measurement noise)

  Complicates manufacture and commissioning
• State-space vs Input/Output approach:

  Numerical techniques vs well established techniques

  Can model MIMO systems vs popular for SISO systems

  Can model non-linear/constrained systems vs only works for simple systems

  Time domain is more intuitive vs frequency domain is often non-intuative

  Maths and implementation on hardware can be more complex for state-space
• A mathematical model is a derivation of the differential/difference equations that govern a physical system. It is an abstract model used to predict the outcome of a system in the real world.
• Mathematical models are described by:

  ODEs - ordinary differential equations (this course)

  PDEs - partial differential equations e.g. wave-diffusion equation

  DAEs - differential-algebraic equations e.g. resistor inductor circuit

  PDAEs - e.g. Navier-Stokes equations
• Linear time invariant (LTI) system:

  Linear:

  a*x_1 + b*x_2 ➡️ a*y_1 + b*y_2

  Time invariant:

  Characteristics are independent of time. A time shift in the input will result in an identical shift in the output.
• Characterise the LTI state-space system equation:

  dx/dy = Ax +Bu

  y = Cx + Du
• Each term in the LTI state-space system equation:

  x - the state vector

  u - the input vector

  y - the output vector

  A matrix - the underlying dynamics

  B matrix - how the inputs enter into the system

  C matrix - how the states are combined to get the outputs

  D matrix - can allow the inputs to bypass the system and feed forward to the output
• What is the feedback diagram of state-space systems?

  (See system diagram from lecture notes)
• Discrete-time state-space system vs continuous time system:

  Numerical simulation vs physical system

  dynamics described as a function of [k] vs as a function of time (t)

  Requires samples vs continuous signal
• In a discrete time system what does dx(t)/dy(t) become?

  x[k+1]
• A non-linear system has a vector field in the form:

  dx(t)/dy(t) = f(x, u)

  y(t) = h(x, u)
• Common non-linearities:

  Saturation - e.g. the limits of a throttle or brake

  Dead-zone - e.g. the loss of friction on tyres on an icy patch in a road

  Quantisation - e.g. digitally stored information
• Why would we want to linearise a model?

  To make control easier when we wish to maintain the system about a fixed operating point
• What are equilibrium points and how can they be calculated?

  x_e is an equilibrium point if:

  x(0) = x_e ➡️ x(t) = x_e for all t≥0

  (the system is at rest)

  Compute equilibrium points by solving:

  f(x_e , u_e) = 0

  (set the derivative equal to zero)
• What are the limitations of using a linearised model?

  The approximation gets less accurate the further from the equilibrium point the initial conditions are