iaac international symposium in systems & control, 7-8 october 2013, technion, haifa, israel p-o...

IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel

P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

Constrained control of uncertain linear time-invariant systems: an interpolation based approach

Per-Olof Gutman

Abstract: In this paper, a novel approach to control uncertain discrete-time linear time-invariant systems with polytopic state and control constraints is proposed. The main idea is to use interpolation. The control law has an implicit and explicit form. In the implicit form, at each time instant, at most two linear programming problems are solved on-line. In the explicit form, the control law is given as a piecewise a-ne and continuous function of the state. The design method can be seen as a computationally favorable alternative to optimization-based control schemes such as Model Predictive Control. Proofs of recursive feasibility and asymptotic stability are given. Several simulations demonstrate the performance, also in comparison with MPC. Ext-ensions include output feedback, LPV and time-varying systems, and ellipsoidal constraint sets.

Main reference: Hoai-Nam Nguyen, Constrained control of uncertain, time-varying systems: an interpolation based approach, accepted for publication as a Springer book, Lecture Notes in Control and Information Sciences, 2014.



Outline

• Uncertainty and disturbances• Output feedback• Interpolation control via LMI• Interpolation with cost • Example: truck-dolly-trailer• Conclusions• References

• Problem formulation• Constrained control

MPCVertex control

• Interpolation based controlMaximal admissible setControl invariant setImplicit solutionExplicit solution



Problem formulation

under the polytopic state and control constraints

Regulate to the origin

· Extensions- Polytopic uncertainty and polytopic disturbances

- Output feedback, by non-minimal state space representation with xT(k) = [y(k) y(k-1) … u(k-1) ….] - Trajectory tracking - Ellipsoidal constraint sets



Constrained control – an overview

Many solutions, among them• Anti-reset windup, and over-ride control

- Ad-hoc• Optimal control

- Almost always open loop solution• Model Predictive Control

- Implicit: optimal control problem over a finite receding horizon solved at each sampling instant

- Explicit: piecewise affine state feedback control law computed off-line- Extends with complexity to the uncertain plant case

• Vertex control (Gutman and Cwikel, 1986)- Computationally cheap with one LP-problem per sampling instant- Covers the uncertain plant case with no additional complexity- No optimization criterion



Unconstrained LQ incentral orange cell:



on



Advantage • fast on-line computations

Challenges • computation of vertex control values ui at vertices• slow convergence, essentially P-control



or, in a similar way, for any other feedback control



=



It might be desirable to make u as near uo as possible by minimizing c. Let

=

Note: Clearly xv+ CN and xo

+

xv+

time: k+1



, cont’d

Recall:with xv

+ CN and xo+

Since the origin , the vertex control decomposition is feasible: x(k+1) = (k+1)v(k+1), where v CN

Then, clearly, c*(k+1)≤ (k+1) 0, as k , since the vertex control law is asymptotically stabilizing, and hence x(k) reaches in final time where the

stabilizing local control law uo= Kx takes over, with x remaining in .

v

v0

v



Calculation of c*

Non-linear optimization

Linear Programming:



1. Measure the state x(k)2. By LP, compute x(k) as the convex combination of the vertices of

CN, where vi(j) denotes vertex i in

sector j.3. Compute the vertex control component uv(k)= where ui denotes the

pre-computed vertex control value at vertex i.4. Determine the optimal c* by LP.5. 6. What for the next sampling instant k:= k+1

Example:

uo= [-0.5609 -0.9758]x, x , fromunconstrained LQ with Q=I, R=1




Challenges • computation of vertex control

values ui at vertices• slow convergence, essentially

P-control



• The vertex control law is but one of several possible in CN\

• Alternatively, steer the state s.t. maximal contraction w.r.t. CN is achieved, recalling that the Lyaponov function level curves of the vertex control law are shrunken images of CN. Choose u such that the Minkowsky functional



1. Measure the state x(k)2. Determine, by LP, the optimal c*, xv

*, xo*, s.t. x=c*xv

*+(1-c*) xo*

3. Find uv, by LP, as the minimizer of the Minkowski functional4. 5. What for the next sampling instant k:= k+1




Challenges • pre-computation of vertex

control values ui at vertices• slow convergence, essentially

P-control

Comp. time [ms]/sampling interval



Comparison with MPC

Explicit Interpolating Control: 25 cells

Explicit MPC: 97 cells



Interpolation with cost



• A novel interpolation between a global vertex control law and a local control law, that may be locally optimal.

• A method to avoid the explicit computation of the vertex control values.• Like MPC, the new controller tends to get the state away from the

constraints when near them, and satisfy performance specifications when near the set point.

• Proofs of constrained stability for uncertain plants and bounded disturbances, and output feedback

• Like MPC, the new control law is affine over a polyhedral partition of the feasible control invariant set.

• The interpolating control law is considerably simpler than MPC with fewer polyhedral cells in the explicit case; and, in the implicit case, with extremely simple and fast LP-computations whose computational requirements are orders of magnitude less than MPC.

• Extension to LMI based interpolating control with ellipsoidal state constraint sets.

• Extension to interpolating control with quadratic cost.• Extensions to time-varying and LPV systems.

iaac international symposium in systems & control, 7-8 october 2013, technion, haifa, israel p-o...

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