Suboptimal control of linear systems with state and control inequality constraints

Sznaier, Mario; Damborg, M.J.

doi:10.1109/cdc.1987.272491

Cited by 170 publications

(126 citation statements)

References 7 publications

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“…Note that if constraints are not active and a linear model is used, then (8) is equivalent to an infinite-horizon cost if P M is chosen as the solution of an appropriate Riccati equation (Sznaier and Damborg, 1987;Chmielewski and Manousiouthakis, 1996).…”

Section: Mpc Problem Formulation Frommentioning

confidence: 99%

Reverse-Engineering Existing Controllers for MPC Design

Maciejowski

2007

IFAC Proceedings Volumes

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We propose a method for finding the cost function and state estimator to be used in MPC, in order to obtain the same controller as a given low-order controller when constraints are inactive, and hence when the MPC controller is operating within its baseline linear regime. This is a very useful starting point for the development of an MPC controller when a successful existing controller is known, but it is desired to add the constraint-handling capabilities of MPC, and perhaps other functionality such as fault-tolerance.

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Section: Mpc Problem Formulation Frommentioning

confidence: 99%

Reverse-Engineering Existing Controllers for MPC Design

Maciejowski

2007

IFAC Proceedings Volumes

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“…The Hamilton-Jacobi-Bellman HJB equation characterizes the optimal cost function and optimal control action for the problem where N 1 is some horizon, and V 0 = 0. Under the assumptions of feasibility, non-explicit optimal solutions to the HJB 8 are known for the case when N is so large that there are no active or violated constraints beyond this horizon and the unconstrained LQ solution is optimal beyond the horizon Sznaier and Damborg 1987, Chmielewski and Manousiouthakis 1996, Scokaert and Rawlings 1998 . These solutions are based on real-time quadratic programming, where a nitedimensional optimization problem is achieved since V x t + N = x T t + N P x t + N , where P is the solution to the Ricatti equation associated with the unconstrained LQR.…”

Section: Introductionmentioning

confidence: 99%

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

2002

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Optimal feedback solutions to the in nite horizon LQR problem with state and input constraints based on receding horizon real-time quadratic programming are well known. In this paper we develop an explicit solution to the same problem, eliminating the need for realtime optimization. A suboptimal strategy, based on a suboptimal choice of a nite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon, is suggested in order to address the computer memory and processing capacity requirements of the explicit solution. It is shown that the resulting feedback controller is piecewise linear, and the piecewise linear structure is explored and exploited for computational analysis of stability and performance of the suboptimal constrained LQR. The piecewise linear structure can also be exploited for e cient real-time implementation of the controller.

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“…We will discuss more on how to ensure recursive feasibility in the presence of model uncertainties and disturbances in Section 4.2.4. Sznaier and Damborg [1987] proposes a dual mode formulation for linear systems where the local controller κ (x k ) = −K x k and Ψ(x k+N ) = x T k+N P x k+N is the solution to a discrete time version of the unconstrained infinite horizon LQ problem, i.e., The authors show that by iteratively solving (4.5) for an increasing horizon, N , until (4.4f) is satisfied (where T is an invariant set of the system controlled with the local controller, see 4.1) then the solution is also the solution to the constrained LQ problem (4.2). Later versions of this approach have explicitly incorporated (4.4f) as a constraint and use a fix horizon, N .…”

Section: Stabilitymentioning

confidence: 99%

“…If φ(r k − r ) = β r k − r ∞ and β is chosen such that φ(r k − r ) constitutes an exact penalty function, as described in Section 4.2.4, then for all x k where φ(r k − r ) = 0 is a feasible solution to (5.19), we will have V * k =V * k , whereV * k is the solution to the related optimization problem The results from Sznaier and Damborg [1987] show that the dual mode MPC formulation, with terminal state penalty equal to the infinite horizon unconstrained LQ cost and with a local controller that is the unconstrained LQ controller, has the local optimality property, i.e., the finite horizon cost equals that of the infinite horizon LQ problem, V ∞ . From this it is clear that for all x k where V * k =V * k it holds also that V * k = V * ∞ .…”

Section: Examples From the Aeronautical Industry 83mentioning

confidence: 99%

Fighter Aircraft Maneuver Limiting Using MPC : Theory and Application

Simon¹

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Flight control design for modern fighter aircraft is a challenging task. Aircraft are dynamical systems, which naturally contain a variety of constraints and nonlinearities such as, e.g., maximum permissible load factor, angle of attack and control surface deflections. Taking these limitations into account in the design of control systems is becoming increasingly important as the performance and complexity of the aircraft is constantly increasing.The aeronautical industry has traditionally applied feedforward, anti-windup or similar techniques and different ad hoc engineering solutions to handle constraints on the aircraft. However these approaches often rely on engineering experience and insight rather than a theoretical foundation, and can often require a tremendous amount of time to tune.In this thesis we investigate model predictive control as an alternative design tool to handle the constraints that arises in the flight control design.We derive a simple reference tracking MPC algorithm for linear systems that build on the dual mode formulation with guaranteed stability and low complexity suitable for implementation in real time safety critical systems.To reduce the computational burden of nonlinear model predictive control we propose a method to handle the nonlinear constraints, using a set of dynamically generated local inner polytopic approximations. The main benefit of the proposed method is that while computationally cheap it still can guarantee recursive feasibility and convergence.An alternative to deriving MPC algorithms with guaranteed stability properties is to analyze the closed loop stability, post design. Here we focus on deriving a tool based on Mixed Integer Linear Programming for analysis of the closed loop stability and robust stability of linear systems controlled with MPC controllers.To test the performance of model predictive control for a real world example we design and implement a standard MPC controller in the development simulator for the JAS 39 Gripen aircraft at Saab Aeronautics. This part of the thesis focuses on practical and tuning aspects of designing MPC controllers for fighter aircraft. Finally we have compared the MPC design with an alternative approach to maneuver limiting using a command governor. v Populärvetenskaplig sammanfattningStyrsystem i moderna flygplan ska sätta stopp om piloten gör en hastig manöver som äventyrar säkerheten. Speciellt viktigt är det i jaktflygplan där piloten kan tvingas manövrera flygplanet precis på gränsen för vad konstruktionen klarar. Reglermetoder från processindustrin anpassas nu för att passa flyget.En av de viktigaste faktorerna när man konstruerar ett flygplan är att flygplanet ska vara enkelt och säkert att flyga. Därför är det av yttersta vikt att man konstruerar flygplanens styrsystem så att piloten inte kan sätta flygplanet i en sådan situation att säkerheten äventyras. En sådan situation kan vara att piloten styr flygplanet så att det tappar sin lyftkraft och därmed sin förmåga att flyga eller också att kraftig turbulens u...

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Suboptimal control of linear systems with state and control inequality constraints

Cited by 170 publications

References 7 publications

Reverse-Engineering Existing Controllers for MPC Design

Reverse-Engineering Existing Controllers for MPC Design

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

Fighter Aircraft Maneuver Limiting Using MPC : Theory and Application

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