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Problem 3

Consider the following system (with \(n=m=1\) ): \(\dot{x}=x u\), and take the problem of minimizing $$ \int_{0}^{\infty} \frac{1}{8} x^{4}+\frac{1}{2} u^{2} d t $$ among all controls making \(x(t) \rightarrow 0\) as \(t \rightarrow \infty\). Show that there is a solution, given in feedback form by a quadratic feedback \(k(x)=-c x^{2}\), for some \(c>0\).

Expert verified

The feedback control input is given by \(k(x) = -cx^2\) for some positive constant \(c\). By substituting this feedback control into the system and checking its stability, we find that the state \(x(t)\) converges to zero as \(t\) approaches infinity for any \(c > 0\). Furthermore, after evaluating the cost function with the feedback control input, we identify that the cost function is minimized and converges. Thus, there exists a solution in the feedback form given by the quadratic feedback \(k(x) = -cx^2\) for some \(c > 0\).

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Chapter 8

(Infinite gain margin of LQ feedback.) As above, suppose that \(\Pi\) is a positive definite solution of the ARE and that \(Q\) is also positive definite. Pick any \(\rho \in[1 / 2, \infty)\) and let \(F:=-\rho R^{-1} B^{\prime} \Pi\). Show that the closed-loop matrix \(A_{c l}=A+B F\) is Hurwitz. The result in Lemma 5.7.18 is needed in the next proof. This states that the operator $$ \mathcal{L}: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}, \quad \mathcal{L}(X):=M X+X N $$ is invertible if both \(M\) and \(N\) are Hurwitz.

Chapter 8

Suppose that \(L_{f} V(x) \leq 0\) for all \(x\) and that \(\dot{x}=f(x)+G(x) u\) is globally stabilized by \(u=-(\nabla V(x) \cdot G(x))^{\prime}\), as in Proposition 5.9.1. Show that \(u=k(x)\) is an optimal feedback, and \(V\) is the value function, for some suitably chosen cost. (Hint: Let $Q(x):=-L_{f} V(x)+\frac{1}{2} L_{G} V(x)\left(L_{G} V(x)\right)^{\prime}$, which gives (8.64) for which \(R\) ? Use Exercise 8.5.5.)

Chapter 8

Consider the discrete-time time-invariant one dimensional system with \(X=\mathbb{R}\), transitions $$ x^{+}=x+u, $$ and control value space the nonnegative reals: $$ \mathcal{U}=\mathbb{R}_{+} . $$ With \(q(t, x, u):=u^{2}, p(x):=x^{2}, \sigma=0\), and \(\tau=3\), find \(V(t, x)\) for all \(x\) and \(t=0,1,2\) using the dynamic programming technique. Next guess a general formula for arbitrary \(t, \sigma, \tau\) and establish its validity.

Chapter 8

If \(\Sigma\) is a controllable time-invariant linear continuous-time system over \(\mathbb{K}=\mathbb{R}\), then there exists an \(m \times n\) real matrix \(F\) such that \(A+B F\) is a Hurwitz matrix. The above Corollary is, of course, also a consequence of the Pole-Shifting Theorem, which establishes a far stronger result. Next we consider a discretetime analogue; its proof follows the same steps as in the continuous- time case.

Chapter 8

Find the value function \(V\) and the optimal feedback solution for the problem of minimizing \(\int_{0}^{\infty} x^{2}+u^{2} d t\) for the scalar system \(\dot{x}=x^{2}+u\). (Hint: The HJB equation is now an ordinary differential equation, in fact, a quadratic equation on the derivative of \(V\).)

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