Computation of the constrained infinite time linear ... - Semantic Scholar
Computation of the Constrained Infinite Time Linear Quadratic Regulator P. Grieder'
'.
F. Borrelli'. F. Torrisi' and M. Morari'
Abstract This paper presents an efficient algorithin for coiiiputing the solution to the constrained infinite time linear quadratic regulator (CLQR) problem for discrete time systems. The algorithm coinbiiies multi-parametric quadratic programming with reachability analysis to obtain the optiinal piecewise affine (PWA) feedback law. The algorithm reduces the time necessary to compute the PWA solution for tlie CLQR when compared to other approaches. It also determines the minimal finite horizon N,c, such that the constrained finite horizon LQR problem equals tlie CLQR problem for a compact set of states S. The on-line computational effort for the implementation of the CLQR caii be significantly reduced as well, either hy evaluating the PWA solution or hy solving the finite dimensional quadratic program associatcd with tlie CLQR for a horizon of N = Ns.
1 Introduction
Receiit,ly, extensive work has been carried out on colistrained optimal coiitrol of linear, time-invariant systeins. Onc main fociis of research has becn Finite Nonzon Constrained Optimal Control (FHCOC). The solution to FHCOC caii be found by solviiig a quadratic program (QP) of finite dimclision or, as Betnporad et. al. proposed in 131, by e\duatiiig a PIVA state fecdback coiitrol law which was coinputed off-line by solving a nmlti-parametric quadratic program. It is currelit practice to approximate the Infinite Time Constrained LQR (CLQR) problem by receding horizon control. For receding horizon control, an FHCOC problem is solved at each time step, aiid then only the initial value of the optimal input scquence is applied to the plant. The contribution of this paper is a iiovel approach to compute the piecewise affine (PIVA) state feedback solution t o the CLQR. problem. The presented algorithm coinbiiies multi-parametric quadratic programming 13) with reachability analysis to obtain the optimal PWA feedback law. The algorithm significantly reduces the 'Institiit fur Auromatik, E T H - Swiss Federal 1nst.itute of Technology, CH-8092 Ziirich
time necessary to compute the PWA solution for the CLQR when compared to other approaches [l,31. Furthermore, the algorithm computes the finite horizon Ns,such that the coiistrained finite horizon LQR computed for a prediction horizon N q . is equal to the CLQR for the compact set S.
2 Problem Statement a n d Properties
In this paper we will coiisider optimal control problem for discrete-time linear time-invariant systems s(t
(1)
with A E RnX" and B E R"X". Let a ( t ) denote the iiieasured state at time t and z t + k l t denote the p r e dicted state at time t + k given the state at timc t. For brevity we will denote skioas i k . 2.1 Finite-Time Constrained LQR Assume that the states and the inputs of system (1) are subject to tlie following constraints
xEM
R",
U
E
('4
D C R"',
where M and D are polyhedral sets containing the origin, and consider the finite-time constrained LQR problem
+zl~~,zN],
subj. to
zk
tM,
k t 11, . . . , N I , kE[O
ZLk€D, Zk+l
, . . . I
N-11,
= Axk f Buk,
Qr t0,
Q t 0,
R
5 0,
(3a)
(3b)
.(3C) (34 (3e)
Henceforth, we will assume the terminal weight matrix
&, to be equal t,o the solution of the Algebraic Riccati Equation (ARE), which we denote as PAR^. The solution to problem (3) has been studied by Bemporad et al. [3]. We will briefly summarize the main results. By substituting z k = A's(0) Ctz: A ' B U ~ _ ~ prob-~, lem (3) caii be reformulated as
+
i
Jh(x(0)) = z(Oj'Yx(0) + inin U L N U N UN
'Coriesponding Author: E-mail: griedermaut . e e . ethz. ch, Tel. +41 01 632 7313
0-7803-7896-2/03/$17.00 02003 IEEE
+ 1) = A z ( t ) + Bu(t),
s.t.
471 1
GUN 5
w + Es(O),
1
+ z(0)'~U.w, (44
Proceedings 01 the American Control Conference Denver, Colorado June 4-6, 2003
uX-~]'
where the column vector UN [ub,. . . , E WrnN is the optimization vector and 'H, F,Y , G, W , E are easily obtained from &, R,Q f , (1) and (2) (see [3] for details). We denote with X N C R" the set of initial f. states z(0) for which the optimal control problem (4) is feasible, '.e:
X,?
{z(O)E R"131i~E R"",GUN 5 W and Jl;(z(O))< CO}
=
+ Ez(0)
and we denote with U G ( z ( 0 ) )the optimizer of (4). Definition 1 The set of,active constraints AN(z(0)) at point z(0) of problem (4) is defined as
A N ( ~ ( 0 ) )= {i E I I GiU;(z) I = { 1 , 2, . . . , m c ] ,
- Wi -
Eiz
= 01,
where Gi, W; and Ei denote thei-th m w ofthe matrices G , W and E , respectively and G E R m Q x N mN. is the horizon length used to solve (4).
and the polyhedron P, = {z E RnIH,z
'.
F. Borrelli'. F. Torrisi' and M. Morari'
Abstract This paper presents an efficient algorithin for coiiiputing the solution to the constrained infinite time linear quadratic regulator (CLQR) problem for discrete time systems. The algorithm coinbiiies multi-parametric quadratic programming with reachability analysis to obtain the optiinal piecewise affine (PWA) feedback law. The algorithm reduces the time necessary to compute the PWA solution for tlie CLQR when compared to other approaches. It also determines the minimal finite horizon N,c, such that the constrained finite horizon LQR problem equals tlie CLQR problem for a compact set of states S. The on-line computational effort for the implementation of the CLQR caii be significantly reduced as well, either hy evaluating the PWA solution or hy solving the finite dimensional quadratic program associatcd with tlie CLQR for a horizon of N = Ns.
1 Introduction
Receiit,ly, extensive work has been carried out on colistrained optimal coiitrol of linear, time-invariant systeins. Onc main fociis of research has becn Finite Nonzon Constrained Optimal Control (FHCOC). The solution to FHCOC caii be found by solviiig a quadratic program (QP) of finite dimclision or, as Betnporad et. al. proposed in 131, by e\duatiiig a PIVA state fecdback coiitrol law which was coinputed off-line by solving a nmlti-parametric quadratic program. It is currelit practice to approximate the Infinite Time Constrained LQR (CLQR) problem by receding horizon control. For receding horizon control, an FHCOC problem is solved at each time step, aiid then only the initial value of the optimal input scquence is applied to the plant. The contribution of this paper is a iiovel approach to compute the piecewise affine (PIVA) state feedback solution t o the CLQR. problem. The presented algorithm coinbiiies multi-parametric quadratic programming 13) with reachability analysis to obtain the optimal PWA feedback law. The algorithm significantly reduces the 'Institiit fur Auromatik, E T H - Swiss Federal 1nst.itute of Technology, CH-8092 Ziirich
time necessary to compute the PWA solution for the CLQR when compared to other approaches [l,31. Furthermore, the algorithm computes the finite horizon Ns,such that the coiistrained finite horizon LQR computed for a prediction horizon N q . is equal to the CLQR for the compact set S.
2 Problem Statement a n d Properties
In this paper we will coiisider optimal control problem for discrete-time linear time-invariant systems s(t
(1)
with A E RnX" and B E R"X". Let a ( t ) denote the iiieasured state at time t and z t + k l t denote the p r e dicted state at time t + k given the state at timc t. For brevity we will denote skioas i k . 2.1 Finite-Time Constrained LQR Assume that the states and the inputs of system (1) are subject to tlie following constraints
xEM
R",
U
E
('4
D C R"',
where M and D are polyhedral sets containing the origin, and consider the finite-time constrained LQR problem
+zl~~,zN],
subj. to
zk
tM,
k t 11, . . . , N I , kE[O
ZLk€D, Zk+l
, . . . I
N-11,
= Axk f Buk,
Qr t0,
Q t 0,
R
5 0,
(3a)
(3b)
.(3C) (34 (3e)
Henceforth, we will assume the terminal weight matrix
&, to be equal t,o the solution of the Algebraic Riccati Equation (ARE), which we denote as PAR^. The solution to problem (3) has been studied by Bemporad et al. [3]. We will briefly summarize the main results. By substituting z k = A's(0) Ctz: A ' B U ~ _ ~ prob-~, lem (3) caii be reformulated as
+
i
Jh(x(0)) = z(Oj'Yx(0) + inin U L N U N UN
'Coriesponding Author: E-mail: griedermaut . e e . ethz. ch, Tel. +41 01 632 7313
0-7803-7896-2/03/$17.00 02003 IEEE
+ 1) = A z ( t ) + Bu(t),
s.t.
471 1
GUN 5
w + Es(O),
1
+ z(0)'~U.w, (44
Proceedings 01 the American Control Conference Denver, Colorado June 4-6, 2003
uX-~]'
where the column vector UN [ub,. . . , E WrnN is the optimization vector and 'H, F,Y , G, W , E are easily obtained from &, R,Q f , (1) and (2) (see [3] for details). We denote with X N C R" the set of initial f. states z(0) for which the optimal control problem (4) is feasible, '.e:
X,?
{z(O)E R"131i~E R"",GUN 5 W and Jl;(z(O))< CO}
=
+ Ez(0)
and we denote with U G ( z ( 0 ) )the optimizer of (4). Definition 1 The set of,active constraints AN(z(0)) at point z(0) of problem (4) is defined as
A N ( ~ ( 0 ) )= {i E I I GiU;(z) I = { 1 , 2, . . . , m c ] ,
- Wi -
Eiz
= 01,
where Gi, W; and Ei denote thei-th m w ofthe matrices G , W and E , respectively and G E R m Q x N mN. is the horizon length used to solve (4).
and the polyhedron P, = {z E RnIH,z
