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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.)

Short Answer

Expert verified
In this problem, we analyze a feedback control system with given dynamics and stabilization conditions. To prove that \(u=k(x)\) is an optimal feedback and that \(V\) is a value function, we follow these steps: 1. Identify the Hamiltonian and Bellman equations. 2. Define the cost function. 3. Write down the Hamilton-Jacobi-Bellman (HJB) equation. 4. Integrate the HJB equation using the given expression for \(Q(x)\). 5. Determine the optimal control. 6. Show that \(u=k(x)\) is the optimal feedback. 7. Prove \(V\) is the value function under the provided conditions. After implementing these steps, we find that \(u\) is indeed a function of \(x\), like \(k(x)\), and therefore, \(u=k(x)\) can be considered an optimal feedback. Moreover, \(V\) is shown to be a value function under the given conditions, since it is the optimal cost-to-go function when \(u\) is optimal and \(u=k(x)\).
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Step 1: Identify the Hamiltonian and Bellman equations

The optimal control problem is usually analyzed using the Hamiltonian and the Bellman equations. In this case, the Hamiltonian function \( H \) is given by: \[ H(x, u, p) = p \cdot f(x) + u' \cdot Q(x) \] The Bellman function \( V(x) \) in the Hamilton-Jacobi-Bellman (HJB) equation is considered as the optimal cost-to-go function.

Step 2: Define the cost function

The cost function \( J \) for the optimal control problem is defined as \( J(x,u) = \int_{t_0}^{t_f} (u'Ru + Q(x)) dt \). In our case, one needs to find out what \( R \) looks like such that the given conditions of the problem are fulfilled.

Step 3: Write down the Hamilton-Jacobi-Bellman (HJB) equation

The HJB equation for the optimal control problem is: \[ -\frac{\partial V}{\partial t} = \min_u[H(x, u, p)] \]

Step 4: Integrate the HJB equation

In the case of time invariant systems, the HJB equation can be time-integrated. Using the provided expression for \( Q(x) \) and the HJB equation, we get: \[ Q(x) = \min_u\left(-L_{f}V(x) + \frac{1}{2} L_{G} V(x) \left(L_{G} V(x)\right)'\cdot R\right) \]

Step 5: Determine the optimal control

The optimal control \( u^{*} \) minimizes the right-hand side of the HJB equation, which gives the condition: \[ u^{*} = -R^{-1}B^{T}p = -R^{-1}(L_{G}V(x)) \]

Step 6: Show \( u=k(x) \) is the optimal feedback

From step 5, we see that \( u \) is indeed a function of \( x \), just like \( k(x) \), thus \( u=k(x) \) can be regarded as an optimal feedback.

Step 7: Prove \( V \) is the value function

From the HJB equation, \( V \) is the optimal cost-to-go function given that \( u \) is optimal, given that \( u = k(x) \). Thus, \( V \) is indeed a value function under the given conditions.

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Most popular questions from this chapter

Chapter 8

Show that, if \((A, C)\) is observable, then $\Pi:=\lim _{t \rightarrow \infty} \Pi(t)$ exists and that it satisfies the dual algebraic Riccati equation $$ A \Pi+\Pi A^{\prime}-\Pi C^{\prime} Q C \Pi+B R^{-1} B^{\prime}=0 . $$

Chapter 8

Consider the case when \(B=0, Q=I\), and \(S\) approaches zero. Show that the formulas for least-squares observation in Section \(6.3\) can be recovered from the results in this section. (Hint: The equation for \(\widetilde{P}\) can be solved with final condition zero, and its solution at the initial time can be expressed directly in terms of the Wronskian \(W\).)

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

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 \(c>0\).

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.

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