So far, we have been focusing on Linear Control Theory, where the governing differential equation of the modelled system is described by: \[ \begin{align*} \begin{split} \mathbf{\dot{x}}(t) &= \mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t)\\ \mathbf{y}(t) &= \mathbf{C}(t)\mathbf{x}(t)+\mathbf{D}(t)\mathbf{u}(t) \end{split} \tag{1} \end{align*} \] It is no doubt that linear control theory is a mature subject, and a long history of successful applications exist [1]. However, linear systems and linear control theory are not rich enough to describe the wide-range of phenomena existing in nature. A detailed review on the examples of such phenomena (e.g., chaos in logistic map) is introduced in Chapter 1 of [2], [3].
Hence, we now move forward and focus on nonlinear control theory, where the governing differential equation of the modelled system is described by a nonlinear dynamical system: \[ \begin{align*} \begin{split} \mathbf{\dot{x}}(t) &= \mathbf{f}(t, \mathbf{x}(t), \mathbf{u}(t)) \\ \mathbf{y}(t) &= \mathbf{g}(t, \mathbf{x}(t), \mathbf{u}(t)) \end{split} \tag{2} \end{align*} \] In these equations, \(t \in [0,T]\subset \mathbb{R}\)1 and the initial condition is \(\mathbf{x}(0)=\mathbf{x}_0\in\mathbb{R}^{n}\). The maps \(\mathbf{f}(\cdot): [0,T]\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^n\) and \(\mathbf{g}(\cdot): [0,T] \times \mathbb{R}^{n} \times \mathbb{R}^{m}\rightarrow \mathbb{R}^p\) are known.
As we discussed in linear control theory, since \(\mathbf{g}(\cdot)\) is known, the first and foremost step is to find the solution of the following initial value problem: \[ \mathbf{\dot{x}}(t) = \mathbf{f}(t, \mathbf{x}(t), \mathbf{u}(t)), ~~ t\in[0,T], ~~ \mathbf{x}(0)\in\mathbb{R}^n \] However, compared to linear dynamical systems where the solution always exists and is unique, given \(\mathbf{u}(t)\) for \(t\in[0,T]\), one should check whether there exists a solution \(\mathbf{x}(t)\) for \(t\in[0,T]\), and whether it is unique. We show some examples to clarify this point.
Consider the following scalar nonlinear differential equation (Chapter 1, Equation 1.6 of [3], Chapter 1, Example 16 of [4]): \[ \dot{x}(t) = -\text{sgn}(x(t))= \begin{cases} -1 ~~~~ \text{for } x(t)\ge0 \\ \phantom{-}1 ~~~~ \text{for } x(t)<0 \end{cases}, ~~ t\in[0,\infty), ~~x(0)=0 \] In this equation, \(\text{sgn}(\cdot):\mathbb{R}\rightarrow \{-1,1\}\) is a sign function. It is clear that since \(\text{sgn}(\cdot)\) is discontinuous, there exists no continuous differentiable solution of \(x(t)\) in \(t\in[0,\infty]\).
Consider the following scalar nonlinear differential equation (Chapter 1, Equation 1.7 of [3], Similar examples are Chapter 1, Example 18 of [4], Chapter 3 Equation 3.3 of [5]): \[ \dot{x}(t) = 3x(t)^{2/3}, ~~ t\in[0,\infty),~~ x(0)=0 \] It is clear that \(x(t)=0\) for \(t\in[0,\infty)\) is a solution. However, using separation of variables: \[ \int_{0}^{x(t)} x^{-2/3}dx = 3 \int_0^{t} d\tau, ~~~~~~~~ x(t) = t^3 \] Hence, there are two solutions \(x(t)=0\) and \(x(t)=t^{3}\) for \(t\in[0, \infty)\).
Note that \(T\) can go to infinity, which therefore sets the domain as \(t\in[0,\infty)\). ↩︎