Stability and stabilization of hybrid systemsMikael Johansson
School of Electrical Engineering
KTH
Stability and stabilization of hybrid systems
Mikael Johansson
School of Electrical Engineering
KTH
Part III – Examples
Constrained control via min-max selectors
Substrate feeding control
Automatic gear-box control
A simple relay system
Constrained control via min-max selectors
Common “pre-HYCON” approach for constrained control
Aim: tracking primary variable (y), while
keeping secondary variable (z) within limits
Numerical example
Specific example with
and proportional constraint controllers.
Control without constraint handling Control with constraint handling
A loop transformation
Linear system interconnected with 3-input/1-output nonlinearity
Loop transformation reduces dimension of nonlinearity by one:
still, few techniques apply to such systems
(e.g. small gain and LDI do not work)
Stability analysis
However, nonlinearity (and hence system) is piecewise linear:
LMI computations return quadratic Lyapunov function
(but S-procedure needed)
Part III – Examples
Constrained control via min-max selectors
Substrate feeding control
Automatic gear-box control
A simple relay system
Fed-batch cultivation of E. coli
[Velut, 2005]
Recombinant (genetically modified) E. coli bacteria used to produce proteins.
Bioreactor control: Add feed (nutrition) and oxygen to maximize cell growth.
Fed-batch: feed added continuously, at limiting rate
Control objective
Objective: maximize feed rate while ensuring that
oxygen level does not drop too low (acetate production, inhibited growth)
glucose is not in excess (“overflow metabolism”)
Probing control
Control strategy: increase feed while no acetate formed, decrease otherwise
Acetate formation detected by probing:
add pulse in feed, observe if oxygen consumed
A piecewise linear abstraction
Simplified model of reactor dynamics
where is a piecewise linear function
Integrating the response over a pulse period,
we find the discrete-time model
Piecewise linear if uk is a linear in x.
Control strategy
Assume a linear integral control
fixed length of probing cycle T and probing pulse T-Tc
To model saturation in glucose uptake, consider
This results in a piecewise linear systems with three regions
(why not two?)
Control objective is now to drive system towards saturation.
Control to saturation
The formulation in Lecture 2 does not return any feasible solution
integrator dynamics in unbounded regions not exponentially stable
Two potential approaches:
Prove convergence for initial values within (large but bounded) region (can be done by adding S-procedure terms)
Remove implicit equality constraints by state-transformation (more satisfying, but more complex; see Velut)
With modifications, stability can (often) be proven VIA pwq Lyapunov fncs.
Numerical results
Stability regions for one specific problem instance (reactor parameters)
red dots bound region where stability can be established numerically
shaded regions are shown to be unstable (via local analysis)
Performance analysis
Stability often not enough with stability – would like to optimize performance
for example, the ability to track time-varying saturation level
Can compute bound on performance
for all reference trajectories r[k] via LMI computations.
Note: typically large system descriptions…
Numerical example
Simulations for specific r[k] for all rate-limited references
Parameter contours suggest optimal parameters
Tuning rules
Similar behavior observed for various parameter values of the process.
Based on this observation, Velut suggests the following tuning rules
where (t) is the unit step response of the linear dynamics.
Part III – Examples
Constrained control via min-max selectors
Substrate feeding control
Automatic gear-box control
A simple relay system
A simple model for car dynamics
[Pettersson, 1999]
Simple model:
Inputs: motor torque T and road incline ; output
where is the discrete input, determined by the current gear
To emphasize this dependence, we write
Gear-switching
Gear-switching strategy:
Can be represented by hybrid automaton with four discrete states
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