Heuristically enhanced feedback control of constrained discrete-time linear systems

Abstract

A theoretical framework to analyze the effects of using on-line optimization in the feedback loop is developed and applied to obtain a suboptimal controller guaranteed to yield asymptotically stable systems.Key Words--Computer control; constrained systems; dynamic programming; feedback control; on-line operation; optimization; stability; suboptimal control; trees.Abstract--Recent advances in computer technology have spurred new interest in the use of feedback controllers based upon on-line minimization for the control of constrained linear systems. Still the use of computers in the feedback loop has been hampered by the fact that the amount of time available for computation in most sampled data systems is not enough to achieve a complete solution using conventional algorithms. Several "ad hoc" techniques have been proposed, but their applicability is restricted by the lack of supporting theory. In this paper we present a theoretical framework to analyze the stability of the closed-loop system resulting from the use of on-line optimization in the feedback loop. Using these results we show that a suboptimal algorithm, based upon the use of heuristic search techniques, yields asymptotically stable systems, provided that enough computation power is available to solve at each sampling interval an optimization problem considerably simpler than the original. The controller presented in this paper is valuable for situations where the customary approaches of using Pontryagin's minimum principle or storing a family of extremal curves are not applicable due to limitations in the computational resources available.

State-Space Quantization Design for the Suboptimal Control of Constrained Systems Using Neuromorphic Controllers

“…then, the system (1) subject to the constraints (2) is control quantized null controllable in g. The proof of the theorem is a straightforward extension of corollary 7-1 in [2].…”

“…7: The system (1) is State Quantized Null Controllable in a region C c C if, for any open set 0 c C containing the origin in its interior, there exists a number s3(C, 0) e R such that for all the quantizations xJ of C with s > Sr and for any initial condition ZL E C, there exist a finite number n, a sequence of admissible controls uy E Q, k = 1,2.. .n, a point z, E 0, and a sequence {g.j, Z& E C such that .k-^, Following previous work in this area [1][2][3], we proceed now to introduce a restriction on the class of constraints allowed in our problem. As it will become apparent latter, the introduction of this restriction, termed the constraint qualification hypothesis, while not affecting significantly the number of real world problems that can be handled by our formalism, introduces more structure into the problenm This additional structure wil become essential in showing constrained controllability.…”

“…In this case, of considerable practical importance for applications ranging from aerospace to process control, the problem of steering the system from a given initial condition to a desired target set usually does not admiit a linear feedback control law as a solution. Therefore, control engineers have to resort to a number of schemas that include (in increasing order of sophistication) switching between several linear controllers, non-linear controllers and online optimization based techniques [1][2][3]. Clearly, a trainable, Neural-Net based controller could provide a welcomed addition to the handful of techniques available for dealing with constrained systems, with the added bonus that such a controller could achieve good performance even in the face of poor or minimal modeling.…”

Explicit sub-optimal linear quadratic regulation with state and input constraints

“…The result follows immediately from the observation that constraining the input will never decrease the value of the optimal cost function, see also Sznaier and Damborg 1990 where a similar result is provided. N. Note that the minimum k x; g; h need not be unique.…”

Model predictive control: Terminal region and terminal weighting matrix

“…Besides terminal penalty techniques, stability-enforcing constraints are often used to develop stable MPC algorithms, where stability is achieved by adding extra constraints into the on-line optimization problem in order to enforce the state to contract to the origin in each step; see e.g. [21][22][23]. Recently, the stability of MPC has been established by utilizing an appropriately designed backup controller and coupling it with MPC implementation [24] as well as Lyapunov-based predictive control designs that guarantee feasibility from an explicitly characterized set (not restricted to the terminal region) of initial conditions [25], and these results were extended to the cases where state constraints [26], uncertainty [27], and rate constraints [28] need to be considered.…”