1. (a) Let P ⊆Rⁿ be a convex set, and let x ε Rn with T_P (x) 6= Φ. Prove that T_P (x) is a convex cone.

(b) Let P = [0, 1] × [0, 1]. Find T_P (x) for every x ε R2.

2. Let S ⊆ R⁴ be the standard orthogonal simplex. Let a control system on S be defined by

Determine ES and OS for this system.

3. The following system arises in the investigation of a standard problem on the regulation of water temperature in a system of interconnected tanks:

The following safety constraints are given:
• 25 ≤ x1 ≤ 80,
• -0.6 ≤ x2 ≤ 1.4,
• x2 ≤ x1/50 if x2 ≥ 0,
• x2 ≥ -x1/100 if x2 ≤ 0.

The control objective is the following: For every initial condition that satisfies the safety constraints, there needs to exist a time T ≥ 0 such that 50 ≤ x1(t) ≤ 60 for all times t ≥ T. Do the following:

(i) Given the above safety constraints, draw the appropriate state space.

(ii) In the remainder of this problem, we will be using the following state space:

Notice that this state space is slightly different from the one you obtained in part (i). Among other things, the vertex (80, 1.4) has been slightly adjusted to (76, 1.36). By investigating the feasibility of the invariance conditions at (80, 1.4) and (76, 1.36), explain why such an adjustment was useful.

(iii) We will be using the following triangulation of the state space from part (ii):

It is known that affine controllers guaranteeing the desired exit behaviour, represented by arrows in the figure above, exist on simplices S5, . . . , S13. Having in mind the control objective of this system, think of a desired exit or stabilization behaviour on simplices S1, S2, S3 and S4. Find affine controllers which enable this behaviour on these simplices. The controllers are allowed to be different on each simplex, and there is no continuity requirement between them. (Warning: not every stabilization/exit behaviour is feasible! If the first idea doesn't work, try a different desired behaviour.)

(iv) The controllers achieving desired behaviour on simplices S5, . . . , S13 are given by:

uS5 = [-45.6 2630]x - 492,
uS6 = [-16.742 - 255.6]x + 1239.426,
uS7 = [-3.95 - 42.38]x + 216.006,
uS8 = [-3.95 - 176.55]x + 216.006,
uS9 = [-2.7 - 134.87]x + 140.998,
uS10 = [-4.45 - 134.87]x + 246,
uS11 = [-1.8398 - 53.88]x + 89.999,
uS12 = [0.90498 - 2.879]x - 21.7496,
uS13 = [0.14 73.62]x + 16.5.

Using these controllers and the controllers you developed in (iii), show that the system achieves the desired behaviour while staying within the safety constraints. In particular, simulate in MATLAB the closed-loop trajectories of the system by choosing each vertex of the polytopic state space as an initial condition. Plot all these trajectories on a phase portrait that also shows the triangulation.

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