# Set-Point Control of a Spatially Distributed Buck Converter
## 4 Discrepancy-Based Control  
Discrepancy-based control [22] is a control approach for distributed parameter systems. The main idea is the investigation of the control problem in a generalized stability setting, stability with respect to two discrepancies [20; 21]. In the following, the main concepts according to [22] are presented.  
Consider a dynamical system with a solution \(\varphi(\cdot,t)\) and an equilibrium at zero \(\varphi_{0}=0\). The discrepancy \(\rho(\varphi(\cdot,t),t)\) is than the generalized distance between the solution \(\varphi(\cdot,t)\) and the equilibrium \(\varphi_{0}\).  
**Definition 1** (Discrepancy [21]).: _A discrepancy is a real valued functional \(\rho=\rho(\varphi(\cdot,t),t)\) with the following properties_  
1. \(\rho(\varphi,t)\geq 0\)_,_
2. \(\rho(0,t)=0\)_,_
3. _For an arbitrary process_ \(\varphi=\varphi(\cdot,t)\) _the real valued functional_ \(\rho(\varphi(\cdot,t),t)\) _is continuous with respect to_ \(t\)_._  
Obviously, the discrepancy is a generalization of more traditional distance measures as the \(L_{p}\)- and the \(L_{\infty}\)- norm.  
To account for deviations of the initial state \(\varphi(\cdot,0)\) from the equilibrium \(\varphi_{0}\), a second time independent discrepancy \(\rho_{0}\) fulfilling some continuity condition can be used. Stability with respect to two discrepancies can then be defined as follows.  
**Definition 2** (Stability with respect to two discrepancies \(\rho\) and \(\rho_{0}\)[21]).: _The equilibrium \(\varphi_{0}=0\) is stable in the sense of Lyapunov with respect to the two discrepancies \(\rho\) and \(\rho_{0}\) for all \(t\geq t_{0}\) if for every \(\varepsilon>0\) and \(t_{0}\geq 0\) there exists a \(\delta=\delta(\varepsilon,t_{0})>0\), such that for every process \(\varphi(\cdot,t)\) with \(\rho_{0}<\delta(\varepsilon,t_{0})\) it follows that \(\rho<\varepsilon\) for all \(t\geq t_{0}\). If in addition \(\lim_{t\to\infty}\rho=0\), then the equilibrium \(\varphi_{0}\) is called asymptotically stable in the sense of Lyapunov with respect to the two discrepancies \(\rho\) and \(\rho_{0}\)._  
To introduce an according Lyapunov functional the notion of positive definiteness with respect to a discrepancy has to be introduced.  
**Definition 3** (Positive definiteness with respect to a discrepancy \(\rho\)[21]).: _A functional \(V=V[\varphi,t]\) is positive definite with respect to a discrepancy \(\rho\), if \(V\geq 0\) and \(V[0,t]=0\) for all \(\varphi\) with \(\rho(\varphi,t)<\infty\) and for every \(\varepsilon>0\) there exists a \(\delta=\delta(\varepsilon)>0\), such that \(V\geq\delta(\varepsilon)\) for all \(\varphi\) with \(\rho[\varphi,t]\geq\varepsilon\)._  
The basis for the discrepancy-based control design, is given by the following theorem providing a connection between (asymptotic) stability with respect to two discrepancies and the existence of an according Lyapunov functional \(V\).  
**Theorem 1** (Lyapunov functional [21]).: _The process \(\varphi\) with the equilibrium \(\varphi_{0}=0\) is stable with respect to the two discrepancies \(\rho\) and \(\rho_{0}\) if, and only if, there exists a functional \(V=V[\varphi,t]\) positive definite with respect to the discrepancy \(\rho\), continuous at time \(t=t_{0}\) with respect to \(\rho_{0}\) at \(\rho_{0}=0\) and not increasing along the process \(\varphi\), i.e., \(V\leq 0\). The process is asymptotically stable if in addition \(\lim_{t\to\infty}V=0\)._As will be shown in the next section, the main step for a discrepancy-based control design is to define an appropriate discrepancy, which allows for a simple Lyapunov-based control design and incorporates process knowledge.