Quadratic Control Problem Research Articles

Output feedback gains for a linear-discrete stochastic control problem

For the finite-horizon linear-discrete quadratic stochastic control problem, the control is restricted to be a memoryless linear transformation of the measurement. The two-point boundary value problem that specifies the feedback gain matrices is derived, and an algorithm for solving it is given. An example is solved comparing the cost of the suboptimal control to the optimal control.

An approximation theory for oscillations of differential equations

In an earlier paper we gave an approximation theory of focal points and focal intervals. The fundamental purpose of this paper is to show that the ideas and methods of that paper can be used to give an approximation theory for oscillation for linear selfadjoint differential (and integral-differential) equations. These methods follow from the theory of quadratic forms given by Hestenes. In §1 we give the preliminaries needed for this paper. In §2 we define oscillation for quadratic control problems and discuss their connection with differential equations. In §3 we give our main approximation results relating the oscillation points λ n ( σ 0 ) ( n = i , 2 , 3 , ⋯ ) {\lambda _n}({\sigma _0})(n = i,2,3, \cdots ) for a given problem to the oscillation points λ n ( σ ) ( n = 1 , 2 , 3 , ⋯ ) {\lambda _n}(\sigma )(n = 1,2,3, \cdots ) of an approximating problem. The element σ \sigma belongs to a metric space ( Σ , d ) (\Sigma ,d) . The main result is to show that the n n th oscillation point λ n ( σ ) {\lambda _n}(\sigma ) is a continuous function of σ ( n = i , 2 , 3 , ⋯ ) \sigma (n = i,2,3, \cdots ) . For completeness in §4 we present an example for fourth order differential equations where the approximation is by discrete problems. Thus oscillation points can be “easily” computed by numerical algorithms.

Dynamic programming for stochastic control of discrete systems

This paper treats the general discrete-time linear quadratic stochastic control problem. This problem is solved in two steps. Dynamic programming is used to obtain a solution to the stochastic control problem in which perfect measurements of the state are available. Then the stochastic control problem in which only noisy measurements of a linear operator on the state are available is converted into a new stochastic control problem in which perfect measurements of the state are available. This conversion is based upon Kalman filter theory and is valid whenever the disturbances and measurement noises are Gaussian.

Error Bounds of High Order Accuracy for the State Regulator Problem via Piecewise Polynomial Approximations

Consider the linear quadratic cost control problem $\dot x = A(t)x + B(t)u$, $x(0) = x_0 $,with a cost functional $J[u] = \frac{1}{2}\int_0^T {[\langle {x,Q(t)x} \rangle + \langle {u,R(t)u} \rangle ]dt} $. Let S be a suitable space of piecewise cubic polynomials on a mesh of norm h on the interval $[0,T]$. Then it is shown that a so-called Ritz–Trefftz method for minimizing $J[ \cdot ]$ over S leads to an approximation to $J[ \cdot ]$ of order $O(h^7 )$. Further, a computable error bound can be exhibited. It is also shown that the computed pair $(\bar u,\bar x)$ converges to the optimal pair $(u^ * ,x^ * )$ with order $O(h^3 )$. Similar statements are made for piecewise polynomial approximation of arbitrary positive order.

Direct method approximation to the state regulator control problem using a Ritz-Trefftz suboptimal control

The linear quadratic cost control problem \dot{x}(t) = A(t)x(t) + B(t)u(t) x(0) = x_{0} with a cost functional J[u] = \frac{1}{2} \int\min{0}\max{T} [\langlex, Q(t)x\rangle + \langleu, R(t)u\rangle] dt is considered, supposing S is a suitable space of piecewise cubic polynominals on a mesh of norm h on the interval [0, T] . Then a Ritz type algorithm is developed for minimizing J [\cdotp] over S . The authors have previously discussed [3] certain convergence properties of the algorithm. Here the algorithm is discussed in a form suitable for real-time implementation and additional convergence criteria are presented. In [3] it was shown that the Ritz-Treffiz suboptimal control \bar{u} converges to the optimal control u\ast with order 0(h^{3}) . If x_{\bar{u}} is the trajectory generated by \bar{u} , then it is shown that x_{\bar{u}} approximates the optimal trajectory x\ast to 0(h^{3}) . Finally, it is shown that J[\bar{u}] approximates J[u\ast] to order 0(h^{6}) . The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.

Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses

Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness

Quadratic control problem with an energy constraint; approximate solutions

The quadratic-error regulator problem with a control-energy constraint has been investigated from the viewpoint of having limited control energy available or as an indirect method of limiting the maximum control amplitude. The solution of this problem requires iteration on a parameter λ n + 1 with the required value of λ n + 1 being a function of the initial state. Thus the result is not as straightforward to apply as the well known Kalman result where no constraints are imposed; however, it is much simpler than the problem where control amplitude constraints are applied directly for which case one must iterate on an n dimensional vector. Also, for the systems studied it was found that λ n + 1 had a regular behavior as a function of initial state and that this behavior could be approximated by empirical equations. Therefore, an approximate control law could be computed for each initial state without iterating yielding real-time solutions. Three examples were considered in the numerical evaluation portion of the study. In addition to revealing the orderly behavior of the required value of λ n + 1 as a function of initial state, the numerical results also indicated that constraining control energy was quite effective in constraining maximum control amplitude.