understanding solution of state equation

Question

Could anyone help me to understand that how the solution of $(1)$ is written as in $(2)$?

I know that $\dot x=Ax+Bu$ is $x(t)=e^{At}x(0)+\int_{0}^{t} e^{A(t-s)} Bu(s)ds$

Edit: After the pauls answer: Here I will have solutions in each subintervals as follows?

$x(t)=e^{A_{i_0}t}x(t_0)+\int_{t_0}^{t} e^{A_{i_0}(t-s)} B_{i_0}u_{i_0}(s)ds$

$x(t)=e^{A_{i_1}t}x(t_1)+\int_{t_1}^{t} e^{A_{i_1}(t-s)} B_{i_1}u_{i_1}(s)ds$

$x(t)=e^{A_{i_2}t}x(t_2)+\int_{t_2}^{t} e^{A_{i_2}(t-s)} B_{i_2}u_{i_2}(s)ds$

$\vdots$

$x(t)=e^{A_{i_s}t}x(t_s)+\int_{t_s}^{t} e^{A_{i_s}(t-s)} B_{i_s}u_{i_s}(s)ds$

Answer 1

It is not surprising you were confused by this. The text can't seem to follow a consistent symbolism, changing how things are denoted constantly, and explaining the changes only after using the new symbols are used.

What we have is (introduced in what I hope is a more clear order):

• $M = \{1, 2, 3, ..., m\}$ represents a collection of switch values.
• $t_0 < t_1 < t_2 < ... < t_s < t_{s+1} = t_f$ are a sequence of real numbers, where $s$ is some fixed integer. $[t_0, t_f]$ is the time interval the problem covers, which is partitioned by the sequence into sub-intervals $[t_k, t_{k+1}]$ for $0 \le k \le s$.
• There is also given a sequence of switch values $i_k \in M, 0 \le k \le s$.
• From the partition and the switch sequence we define $\sigma : [t_0, t_f] \to M$ by $$\sigma(t) = \begin{cases}i_0& t_0\le t < t_1\\i_1& t_1\le t < t_2\\\vdots\\i_s& t_s\le t < t_{s+1}\end{cases}$$$\sigma$ is called the switching path.
• Each switch value $i\in M$ has associated to it a state space of dimension $r_i$, represented by $\Bbb R^{r_i}$ (the text uses $\frak R$ to represent the real numbers, but $\Bbb R$ is conventional). Each $i$ also has a $n \times n$ matrix $A_i$ and an $n\times r_i$ matrix $B_i$, and an "input function" $u_i : [t_0, t_f] \to \Bbb R^{r_i}$. (As far as the math here is concerned, we could simplify this by defining $U_i : [t_0, t_f] \to \Bbb R^n : t \to B_iu_i(t)$. Then we only need to consider $U_i$ instead of both $B_i$ and $u_i$.)
• Finally we have the state vector function $x : [t_0, t_f] \to \Bbb R^n$, which satisfies the differential equation $$\dot x(t) = A_{\sigma(t)}x(t) + B_{\sigma(t)}u_{\sigma(t)}(t) = A_{\sigma(t)}x(t) + U_{\sigma(t)}(t)$$ Since $\sigma$ is constant on the partition intervals, this breaks down into a set of separate differential equations on each of the partitions: $$\dot x(t) = A_{i_k}x(t) + U_{i_k}(t)\quad t \in [t_k, t_{k+1})$$

If we just look at a single partition interval, the solution is $$x(t) = e^{A_{i_k}(t-t_k)}V_k + \int_{t_k}^te^{A_{i_k}(t-\tau)}U_{i_k}(\tau)\,d\tau$$ for some constant vector $V_k$ (my notation, not his). Based on criteria I do not find in this snippet of text, apparently the constant vector $V_k$ is given by $$V_k = Q_k\left(e^{A_{i_0}(t_1-t_0)}x_0 + \int_{t_0}^{t_1}e^{A_{i_0}(t_1-\tau)}U_{i_0}(\tau)\,d\tau\right) + ...$$where$$Q_k = e^{A_{i_{k-1}}(t_k-t_{k-1})}...e^{A_{i_1}(t_2-t_1)}$$ Presumably, $x_0 = x(t_0)$, but it is unclear to me what the "$...$" in the expression for $V_k$ entails. Since this constant value is deduced from information not given in the snippet, I have no way of determining it.