Third-order Plant Research Articles

A minimal time discrete system

Consider a sampled-data control system with the following sequence of components in the forward path: a sampler with period <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> , a zero-order hold circuit, a linear amplifier with saturation limits ±1, and a plant with transfer function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)=\frac{1}{\Pi\min{i=1}\max{n} (s-\lambda_{i})}.</tex> It is assumed that the poles <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda_{1}, \lambda_{2}, ... , \lambda_{n}</tex> of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)</tex> are real, distinct, and non-positive (a single integral is permissible). The sampler, zero-order hold, and saturating amplifier constrain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> , the forcing function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)</tex> , to be piecewise constant with values between -1 and +1. The forcing function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> is completely defined, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t&gt;0</tex> , by the sequence of numbers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_{1}, f_{2}</tex> , ... , where f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> is the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> during the i'th sampling period. The minimal time regulator problem for the above system can then be stated as follows: Given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)</tex> with an arbitrary set of initial conditions [i.e., the state vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\overrightarrow{c(0)}</tex> defined by its components <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c(0), \dot{c}(0), ... , c^{n-1}(0)</tex> ]; find the forcing function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> [specified by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_{1}, f_{2}, ...</tex> and satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|f_{i}| \leq 1]</tex> , and the corresponding computer in the feedback loop which will bring the system to equilibrium in the minimum number of sampling periods. Any such forcing function will be called an optimal control. The first step is to consider <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{N}'</tex> the set of all initial states <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\overrightarrow{c(0)}</tex> from which the origin can be reached in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> sampling periods or less. From this definition all such states are characterized algebraically and geometrically: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{N}'</tex> is shown to be a convex polyhedron with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 \sum\min{k=1}\max{n} ({N-1}\over{k-1})</tex> vertices. Let R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> be the set of all initial states <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\overrightarrow{c(0)}</tex> from which the origin can be reached in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> sampling periods and no less. Each point of R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> is shown to have a unique canonical representation. The coefficients appearing in the canonical representation suggest an optimal control. To obtain this particular optimal control we define a surface in state space called the critical surface. It is shown that this optimal control will be generated by the following procedure: at the beginning of each sampling period the distance φ from the state of the system to the critical surface is measured along a fixed specified direction; if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi \geq 1</tex> (or ≤ -1) then the forcing function for that sampling period is +1 (or -1); if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|\phi| &lt; 1</tex> , then the forcing function is φ. For a third-order plant it is shown that the critical surface has certain properties which lead to a simple analog computer simulation.