Networked Control (5 CFU) - Project: Six-tanks system
Six-tanks system 01
Consider the system illustrated in the following figure, consisting of a cascade interconnection of six tanks.
The linearized centralized model is
̇ ?h1 = −?h1 + ?1
̇ ?h2= ?h1−?h2+?2
̇ ?h3 = ?h2 +?h4 −?h3
̇ ?h4 = −?h4 + ?3
̇ ?h5 = ?h3 +?h6 −?h5
̇ {?h6 = −?h6 + ?4
where ? = 1 m2 and ? = 1 m2/?. Note that, since the model is linearized around a nominal condition, all the variables of the model above should be regarded as differences with respect to nominal values. We assume that all levels are measurable. Defining ? = [h1, ... , h6]? and ? = [?1, ?2, ?3, ?4]?, the system’s dynamics is described by the model
?̇ = ?? + ??
where the matrices are defined in the corresponding MATLAB file.
Problem:
- Decompose the state and input vectors into subvectors, consistently with the physical description of the system. Obtain the corresponding decomposed model.
- Generate the system matrices (both continuous-time and discrete-time, the latter with a sampling time selected compatibly with the continuous-time dynamics). Perform the following analysis:
- Compute the eigenvalues and the spectral abscissa of the (continuous-time) system. Is it open-loop asymptotically stable?
- Compute the eigenvalues and the spectral radius of the (discrete-time) system. Is it open-loop asymptotically stable?
- For different state-feedback control structures (i.e., centralized, decentralized, and different distributed schemes) perform the following actions
- Compute the continuous-time fixed modes
- Compute the discrete-time fixed modes
- Compute, if possible, the CONTINUOUS-TIME control gains using LMIs to achieve the desired performances. Apply, for better comparison, different criteria for computing the control laws.
- Compute, if possible, the DISCRETE-TIME control gains using LMIs to achieve the desired performances. Apply, for better comparison, different criteria for computing the control laws.
- Analyze the properties of the so-obtained closed-loop systems (e.g., stability, eigenvalues) and compute the closed-loop system trajectories (generated both in continuous-time and in discrete-time) of the water levels starting from a common random initial condition.
