System theory and avanced control - Appunti di Controllo nonlineare: regolazione (Set-Point Stabilization), equazioni di Solvibilità, linearizzazione, Gain Scheduling e controllo integrale

Aggiornato il 13/02/2026

di NoteNinja

Publisher

Vota

Contenuto verificato e approvato dal Team di Esperti di Skuola.net

Regulation problem (set-point stabilization): modello in spazio di stato, uscite misurate vs regolate, controller dinamico/statico, equazioni di solvibilità (plant + regulator), …

Esame System theory and advaced control

Facoltà Ingegneria

Dal corso del Prof. Gradoni Lorenzo

Università Università degli Studi di Bologna

A.A. 2025-2026

95 pagine

Appunti esame

Scarica

Estratto del documento

1.1 The Regulation (or Set-point Stabilization) Problem

Consider the system (all functions are assumed sufficiently smooth)

\[\begin{align*}\dot{x} & = f(x, u) \\x(t) & \in \mathbb{R}^{n_{x}} \, (\text{state}), \, u(t) \in \mathbb{R}^{n_{u}} \, (\text{control}) \\y_{r} & = h_{r}(x) \\y_{r}(t) & \in \mathbb{R}^{n_{r}} \, (\text{regulated output}), \\y_{m} & = h_{m}(x) \\y_{m} & \in \mathbb{R}^{n_{m}} \, (\text{measured output})\end{align*}\]

Let \( y_{r}^{*} \in \mathbb{R}^{n_{r}} \) be a desired value (set-point) for \( y_{r}(t) \)

Problem: Design a controller of the form

\[\begin{align*}\dot{\xi} & = g(\xi, y_{m}) \\\xi(t) & \in \mathbb{R}^{n_{s}} (\xi - \text{controller's state}) \\u & = \gamma(\xi, y_{m})\end{align*}\]

such that the closed-loop system

\[\begin{align*}\dot{x} & = f(x, \gamma(\xi, h_{m}(x))) \\\dot{\xi} & = g(\xi, h_{m}(x)) \\y_{r} & = h_{r}(x)\end{align*}\tag{*}\]

satisfies the following:

There exists an equilibrium point \((x^{*}, \xi^{*}) \in \mathbb{R}^{n_{x}} \times \mathbb{R}^{n_{s}}\) of \((*)\) such that:

\[y_{r}^{*} = h_{r}(x^{*})\]

^{The regulated output \( y_{r} \) equals \( y_{r}^{*} \)} _{at the equilibrium}

The equilibrium \((x^{*}, \xi^{*}) \text{ is (locally/globally/...) asymptotically stable for (*)}

If this problem is solved, then all solutions of the closed-loop system converge to a steady state \((x^{*}, \xi^{*})\) at which \( y_{r} \) attains its desired value \( y_{r}^{*} \)

This problem will be our main focus throughout the module.

Bibliography: H. Khalil, Nonlinear Systems (Chap. 12.2)

1.1 The Regulation (or Set-Point Stabilization) Problem

Consider the system (all functions are assumed sufficiently smooth)

\(\dot{x} = f(x, u)\)
\(y_r = h_r(x)\)
\(y_m = h_m(x)\)

\(x(t) \in \mathbb{R}^{n_x} \, (state)\),\(u(t) \in \mathbb{R}^{n_u} \, (control)\)

\(y_r(t) \in \mathbb{R}^{n_y} \, (regulated \, output)\), \(y_m(t) \in \mathbb{R}^{n_m} \, (measured \, output)\)

variable on which we have performance specifications

we only measure \(y_m(t)\)

Let \(y_r^* \in \mathbb{R}^{n_y}\) be a desired value (set-point) for \(y_r(t)\)

Problem: design a controller of the form

\(\dot{\xi} = g(\xi, y_m)\)
\(u = \gamma (\xi, y_m)\)

\(\xi(t) \in \mathbb{R}^{n_\xi}\) \( (\xi - controller's \, state)\)

A subcase is the static controller:

\(u = \gamma \left( y_m \right)\)

such that the closed-loop system

\(\dot{x} = f \left( x, \gamma \left( \xi, h_m(x) \right) \right)\)
\(\dot{\xi} = g\left( \xi, h_m(x) \right)\)
\(y_r = h_r(x)\)

\((\ast)\)

satisfies the following:

There exists an equilibrium point \((x^*, \xi^*) \in \mathbb{R}^{n_x} \times \mathbb{R}^{n_\xi}\) of \((\ast)\) such that:
- \(y_r^* = h_r \left(x^*\right)\)

Anteprima

Vedrai una selezione di 20 pagine su 95