Example_4_0

Example 4.0:

Following the model of a physical component

$$\begin{array}{ll} \dot{x} = f(x,u), & C := \{ x \mid x \in \mathbf{R}^{n}\}, \\\ x^+ = g(x) = \emptyset, \quad & D := \emptyset, \\\ y = h(x,u), \end{array}$$

a linear time-invariant model of the physical component is defined by

$$ \begin{array}{l} f(x,u) = f_P(x,u)= A_P x + B_P u, \\ h(x,u) = h(x,u) = M_P x + N_P u \end{array} $$

where $A_P$ , $B_P$ , $M_P$ , and $N_P$ are matrices of appropriate dimensions. State and input constraints can directly be embedded into the set $C_P$ . For example, the constraint that $x$ has all of its components nonnegative and that $u$ has its components with norm less or equal than one is captured by

$$\begin{array}{ll} C_P = \{(x,u)\in\mathbf{R}^{n_P}\times \mathbf{R}^{m_P}\mid x_i \geq 0 \ \forall i \in \{1,2,\ldots,n_P\}\} \\ \quad\qquad {} \cap \{(x,u)\in\mathbf{R}^{n_P}\times \mathbf{R}^{m_P} \mid |u_i| \leq 1 \ \forall i \in \{1,2,\ldots,m_P\}\} \end{array} $$

For example, the evolution of the temperature of a room with a heater can be modeled by a linear-time invariant system with state $x$ denoting the temperature of the room and with input $u = (u_1,u_2)$ , where $u_1$ denotes whether the heater is turned on ( $u_1 = 1$ ) or turned off ( $u_1 = 0$ ) while $u_2$ denotes the temperature outside the room. The evolution of the temperature is given by

$$ \begin{array}{l} \dot{x} = -x + [z_\Delta \quad 1] \left[\begin{array}{c} u_1 \\\ u_2\end{array}\right] \textrm{ when } (x,u)\in C_P = \{(x,u) \in \mathbf{R} \times\mathbf{R}^2 \mid u_1 \in \{0,1\}\} \end{array} $$

where $z_\Delta$ is a constant representing the heater capacity.