ISA Transactions 49 (2010) 114–120
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Robust model predictive control design with input constraintsI Vojtech Veselý ∗ , Danica Rosinová, Martin Foltin Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovicova 3, 81219 Bratislava, Slovak Republic
article
info
Article history: Received 4 May 2009 Received in revised form 14 October 2009 Accepted 14 October 2009 Available online 23 October 2009 Keywords: Model predictive control Robust control Quadratic stability Lyapunov function Polytopic system
abstract The paper addresses the problem of designing a robust output/state model predictive control for linear polytopic systems with input constraints. The new predictive and control horizon model is derived as a linear polytopic system. Lyapunov function approach guarantees the quadratic stability and guaranteed cost for closed-loop system. The invariant set and an algorithm approach similar to Soft Variable-Structure Control (SVSC), ensures input constraints for the model predictive plant control system. In the proposed control scheme, the required on-line computation load is significantly less than in MPC literature, which opens the possibility to use these control design schemes not only for plants with slow dynamics, but also for faster ones. © 2009 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction Model predictive control (MPC) has attracted notable attention in the control of dynamic systems, and gains an important role in control practice. The idea of MPC can be summarized as follows. [1–3]:
• Predict the future behavior of the process state/output over the finite time horizon.
• Compute the future input signals on-line at each step by minimizing a cost function under inequality constraints on the manipulated (control) and/or controlled variables. • Apply only the first vector control variable to the controlled plant and repeat the previous step with new measured input/state/output variables. Therefore, the presence of the plant model is a necessary condition for the development of the predictive control. The success of MPC depends on the degree of precision of the plant model. In practice, modelling real plants inherently includes uncertainties that have to be considered in control design, that is control design procedure has to guarantee stability, performance and robustness properties of closed-loop systems in the whole uncertainty domain. Two typical descriptions of uncertainty, state space polytope and bounded unstructured uncertainty are extensively considered in the field of robust model predictive
I Supported by the Grant Agency of the Slovak Republic under Grant 1/0544/09.
∗
Corresponding author. Tel.: +421 2 60291633. E-mail addresses:
[email protected] (V. Veselý),
[email protected] (D. Rosinová),
[email protected] (M. Foltin).
control. Most of the existing techniques for robust MPC assume measurable state, and apply plant state feedback, or when the state estimator is utilized output feedback is applied. Thus, the present state of the robustness problem in MPC can be summarized as follows: Analysis of robustness properties of MPC. Zafiriou and Marchal [4] have used the contraction properties of MPC to develop necessary sufficient conditions for robust stability of MPC with input and output constraints for SISO systems and the impulse response model. Polak and Yang [5] have analyzed the robust stability of MPC using a contraction constraint on the state. MPC with explicit uncertainty description. Zheng and Morari [6] have presented robust MPC schemes for SISO FIR plants, given uncertainty bounds on the impulse response coefficients. Some MPCs consider additive types of uncertainty (de la Pena et al. [7]) or parametric (structured) type uncertainty using CARIMA model and linear matrix inequality (Bouzouita et al. [8]). In Lovaas et al. [9], for open-loop stable systems having input constraints, the unstructured uncertainty is used. The robust stability can be established by choosing a large value for the control input weighting matrix R in the cost function. The authors proposed a new less conservative stability test for determining a sufficiently large control penalty R using bilinear matrix inequality (BMI). In Casavola et al. [10] robust constrained predictive control of uncertain norm-bounded linear systems is studied. The other technique of constrained tightening to design of robust MPC has been proposed in Kuwata et al. [11]. The above approaches are based on the idea of increasing the robustness of the controller by tightening the constraints of the predicted states. The mixed H2 /H∞ control approach to design of MPC has been proposed by Orukpe et al. [12].
0019-0578/$ – see front matter © 2009 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2009.10.003
V. Veselý et al. / ISA Transactions 49 (2010) 114–120
Robust constrained MPC using linear matrix inequality (LMI) has been proposed by Kothare et al. [13] where the polytopic model or structured feedback uncertainty model have been proposed. The main idea of Kothare et al. [13] is the use of infinite horizon control laws which guarantee robust stability for state feedback. In Ding et al. [14] output feedback robust MPC for systems with both polytopic and bounded uncertainty with input/state constraints is presented. Off-line, it calculates a sequence of output feedback laws based on the state estimators, by solving the LMI optimization problem. On-line, at each sampling time, it chooses an appropriate output feedback law from this sequence. One step ahead prediction robust MPC controller design is proposed in Veselý and Rosinová [15]. The survey of optimal and robust MPC design can be consulted in Mayne et al. [16]. In the MPC approach generally, the control algorithm requires solving constrained optimization problems on-line (in each sampling period). Therefore the on-line computation burden is significant and limits practical applicability of such algorithms to processes with relatively slow dynamics. In this paper, a new MPC scheme for an uncertain polytopic system with constrained control is developed using the model structure introduced in Veselý and Bars [17]. The main contribution of present paper is that all the time demanding computations of output feedback gain matrices are realized off-line (for constrained control and unconstrained control cases). The actual value of the control variable is obtained through simple on-line computation of scalar parameters and respective convex combination of already computed matrix gains. The developed control design scheme employs quadratic Lyapunov stability to guarantee the robustness and performance (guaranteed cost) over the whole uncertainty domain. The paper is organized as follows. A problem formulation and preliminaries on a predictive output/state model as a polytopic system are given in the next section. In Section 3, the approach of robust output feedback predictive controller design using linear matrix inequality is presented. In Section 4, the input constraints are applied to LMI feasible solution. Two examples illustrate the effectiveness of the proposed method in the Section 5. Finally, some conclusions are given. Hereafter, the following notational conventions will be adopted: given a symmetric matrix P = P T ∈ Rn×n , the inequality P > 0 (P ≥ 0) denotes matrix positive definiteness (semi-definiteness). Given two symmetric matrices P, Q , the inequality P > Q indicates that P − Q > 0. The notation x(t + k) will be used to define, at time tk-steps ahead, prediction of a system variable x from time t onwards under a specified initial state and input scenario. I denotes the identity matrix of corresponding dimensions. 2. Problem formulation and preliminaries Consider the following linear discrete-time uncertain system x(t + 1) = Av (α)x(t ) + Bv (α)u(t )
(1)
y(t ) = Cv x(t ) where x(t ) ∈ R , u(t ) ∈ R , y(t ) ∈ R are state, control and output variables of the system, respectively; Av (α), Bv (α) belong to the convex set n
S = {Av (α) ∈ R
( Av (α) =
K X
m
l
, Bv (α) ∈ R } ) K X Aj αj Bv (α) = Bj αj , αj ≥ 0 n×n
n×m
j =1
(2)
j=1 K
j = 1, 2, . . . , K ,
X
αj = 1.
115
Simultaneously with (1), we consider the nominal model of system (1) in the form x(t + 1) = Ao x(t ) + Bo u(t )
y(t ) = Cv x(t )
(3)
where Ao , Bo are any constant matrices from the convex bounded domain S (2). The nominal model (3) will be used for prediction, while (1) is considered as a real plant description providing plant output. Therefore, in the robust controller design, we assume that for time t, output y(t ) is obtained from uncertain model (1), and predicted outputs for time t + 1, . . . , t + N will be obtained from model prediction, where the nominal model (3) is used. The states and outputs of the system (1) for the instant t + k, k = 1, 2, . . . , N are given by
• k=1 x(t + 2) = Ao x(t + 1) + Bo u(t + 1)
= Ao Av (α)x(t ) + Ao Bv (α)u(t ) + Bo u(t + 1) • k=2 x(t + 3) = A2o Av (α)x(t ) + A2o Bv (α)u(t )
+ Ao Bo u(t + 1) + Bo u(t + 2) • for k x(t + k + 1) = Ako Av (α)x(t ) + Ako Bv (α)u(t )
+
k−1 X
Aok−i−1 Bo u(t + 1 + i)
(4)
i =0
and corresponding output is y(t + k) = Cv x(t + k).
(5)
= 0, 1, 2, . . . , N state/output model
Consider a set of k predictions as follows
z (t + 1) = Af (α)z (t ) + Bf (α)v(t ),
yf (t ) = Cf z (t )
(6)
where z (t )T = [x(t )T · · · x(t + N )T ], v(t )T = [u(t )T · · · u(t + Nu )T ]
(7)
yf (t ) = [y(t ) · · · y(t + N ) ] T
T
T
and Bv (α) Ao Bv (α)
Bf (α) =
0 Bo
··· ··· ANo Bv (α) ANo −1 Bo Av (α) 0 ··· Ao Av (α) 0 · · · Af (α) = ··· ··· ··· ANo Av (α) 0 ··· Cv 0 ··· 0 0 Cv · · · 0 Cf = · · · · · · · · · · · · 0 0 · · · Cv
··· ··· ··· ···
0 0 0
(8)
ANo −Nu Bo
0 0
· · · , 0
(9)
where N , Nu are output and control prediction horizons of model predictive control, respectively. Matrix dimensions are Af (α) ∈ Rn(N +1)×n(N +1) , Bf (α) ∈ Rn(N +1)×m(Nu +1) and Cf ∈ Rl(N +1)×n(N +1) . Consider the cost function associated with the system (6) in the form
j =1
Matrices Ai , Bi and Cv are known matrices with constant entries of corresponding dimensions.
J =
∞ X t =0
J (t )
(10)
116
V. Veselý et al. / ISA Transactions 49 (2010) 114–120
where Nu
N
J (t ) =
X
x(t + k)T Qk x(t + k) +
k=0
X
u(t + k)T Rk u(t + k)
k =0
= z (t )T Qz (t ) + v(t )T Rv(t ) Q = blockdiag {Qi }i=0,1,...,N R = blockdiag {Ri }i=0,1,...,Nu .
(11)
The problem studied in this paper can be summarized as follows. Design the robust model predictive controller with output feedback and input constraints in the form
v(t ) = Fyf (t ) = FCf z (t )
(12)
where F T = [F0T · · · FNTu ], Fi = [Fi0 · · · FiN ],
3. Robust model predictive controller design. Quadratic stability
i = 0, 1, 2, . . . , Nu
are the output feedback gain matrices which, for given prediction horizon N and control horizon Nu , guarantee the closed-loop system (13) stability, robustness and guaranteed cost. z (t + 1) = (Af (α) + Bf (α)FCf )z (t ) = Ac (α)z (t ).
(13)
Definition 1. Consider the system (6). If there exists a control law v(t )∗ and a positive scalar J ∗ such that the closed-loop system (13) is stable and the closed-loop cost function (10) value J satisfies J ≤ J ∗ then J ∗ is said to be the guaranteed cost and v(t )∗ is said to be the guaranteed cost control law for the system (6). To guarantee closed-loop stability of uncertain system overall – the whole uncertainty domain – the concept of quadratic stability is frequently used. That is, one Lyapunov function works for the whole uncertainty domain. Experience and analysis has shown that quadratic stability is rather conservative in many cases. Therefore, robust stability with parameter-dependent Lyapunov function P (α) has been introduced [18]. Using the concept of Lyapunov stability, it is possible to formulate the following definition and lemma. Definition 2. System (13) is robustly stable in the convex uncertain domain with parameter-dependent Lyapunov function P (α) if and only if there exists a matrix P (α) = P (α)T > 0 such that Ac (α) P (α)Ac (α) − P (α) < 0. T
(14)
Lemma 1 (Rosinová et al., 2003 [19]). Consider the closed-loop system (13) with control algorithm (12). Control algorithm (12) is the guaranteed cost control law if and only if there exists a positive definite matrix P (α) and matrix F such that the following condition holds Be = z (t )T (Ac (α)T P (α)Ac (α) − P (α) + Q + CfT F T RFCf )z (t )
≤0
(15)
where the first term of (15) 1V (t ) = z (t ) (Ac (α) P (α)Ac (α) − P (α))z (t ) is the first difference of closed-loop system Lyapunov function V (t ) = z (t )T P (α)z (t ). Moreover, summarizing (15) from initial time to to t → ∞, the following inequality is obtained T
− V (to ) + J ≤ 0.
T
Robust MPC controller design which guarantees quadratic stability and guaranteed cost of the closed-loop system is based on (15). Using the Schur complement, formula inequality (15) can be rewritten to the following bilinear matrix inequality (BMI).
−P (α) + Q
FCf Ac (α)
CfT F T −R−1 0
Ac (α)T ≤ 0. 0 −P (α)−1
(18)
For the quadratic stability P (α) = P = P T > 0 in (18). Using a linearization approach with respect to P −1 [20], the following inequality can be derived
− P −1 ≤ Yk−1 (P − Yk )Yk−1 − Yk−1 = lin(−P −1 )
(19)
where Yk , k = 1, 2, . . . in iteration process Yk = P. We can recast bilinear matrix inequality (18) to the linear matrix inequality (LMI) using linearization approach (19). The following LMI is obtained for quadratic stability
−P + Q
FCf Afi + Bfi FCf
CfT F T −R − 1 0
ATfi + CfT F T BTfi ≤ 0 i = 1, 2, . . . , K (20) 0 lin(−P −1 )
where Af (α) =
K X j =1
Afj αj
Bf (α) =
K X
Bfj αj .
j =1
We can conclude that if the LMIs (20) are feasible with respect to % ∗ I > P = P T > 0 and matrix F , then the closed-loop system with control algorithm (12) is quadratically stable with guaranteed cost (17). Note that, due to control horizon strategy, only the first m rows of matrix F are used for real plant control. The other part of matrix F serves for predicted output variables calculation. Parameter-dependent or Polynomial parameter-dependent quadratic stability approach to the design of robust MPC may decrease the conservatism of quadratic stability. In this case, for PDQS we can use the approaches given in [18,22] and for (PPDLF); see [21].
(16) 4. MPC design for input constraints
Definition 1 and (16) imply J ∗ ≤ V (to ).
inequality (15) defines the parameter- dependent quadratic stability (PDQS). A formal approach to choose P (α) for real convex polytopic uncertainty (2) can be found in the references. One of the approaches is to take P (α) = P. In this case, if the solution is feasible, the quadratic stability is obtained. Another apPK PK T proach P (α) = i=1 Pi αi , i=1 αi = 1, Pi = Pi > 0 gives the parameter dependent quadratic stability (PDQS). To decrease the conservatism of PDQS arising from affinely parameter- dependent Lyapunov function (PDLF), more recently, the use of polynomial PDLF (PPDLF) has been proposed in different forms. For more detail, see Ebihara et al. [21].
(17)
Note that, as a receding horizon strategy is used, only u(t ) is sent to the real plant control; control inputs u(t + k), k = 0, 1, 2, . . . , Nu are used for predictive outputs y(t + k) calculation. According to de Oliveira et al. [20], there is no general and systematic way to formally determine P (α) (15) as a function of Ac (α). Such a matrix P (α) is called the parameter-dependent Lyapunov matrix (PDLM) and, for a particular structure of P (α), the
In this section, we propose the off-line calculation of two control gain matrices and, using an analogy to the SVSC approach, Adamy and Flemming [23], we significantly reduce the computational effort for MPC suboptimal control with input constraints. To design model predictive control [23,1] with constraints on input, state and output variables at each sampling time, starting from the current state, an open-loop optimal control problem is solved over the defined finite horizon. The first element of the optimal control sequence is applied to the plant. At the next time
V. Veselý et al. / ISA Transactions 49 (2010) 114–120
step, the computation is repeated with new measured variables. Thus, the implementation of the MPC strategy requires a QP solver for the on-line optimization which still requires significant on-line computational effort, which limits MPC applicability. In our approach the actual output feedback control gain matrix is computed as a convex combination of two gain matrices computed a priori (off-line): one for constrained and one for unconstrained cases, such that both gains guarantee performance and robustness properties of the closed-loop system. This convex combination is determined by a scalar parameter which is updated on-line in each step. Based on this idea, in the following, the algorithm for constrained control algorithm is developed. Consider the system (6) where the control v(t ) is constrained to evolve in the following set
Γ = {v ∈ RmNu : |vi (t )| ≤ Ui , i = 1, . . . , mNu }.
(21)
The aim of this part of paper is to design the stabilizing control output feedback law for system (6) in the form
v(t ) = FCf z (t )
(22)
which guarantees that, for the initial state z0 ∈ Ω (P ) = {z (t ) : z (t )T Pz (t ) ≤ θ } control v(t ) belongs to the set (21) for all t ≥ 0, where θ is a positive real parameter which determines the size of Ω (P ). Furthermore, Ω (P ) should be such that all z (t ) ∈ Ω (P ) provide v(t ) satisfying the relation (21), restricting the values of the control parameters. Moreover, the following ellipsoidal Ljapunov function level set
Ω (P ) = {z (t ) ∈ RnN : z (t )T Pz (t ) ≤ θ }
(23)
can be proven to be a robust positively invariant region with respect to motion of the closed-loop system in the sense of the following definition, [24,25]. Definition 3. A subset So ∈ R(nN ) is said to be positively invariant with respect to the motion of system (6) with control algorithm (22) if, for every initial state, z (0) inside So the trajectory z (t ) remains in So for all t ≥ 0. Consider that vector Fi denotes the i-th row of matrix F and define L(F ) = {z (t ) ∈ R(nN ) : |Fi Cf z (t )| ≤ Ui , i = 1, 2, . . . , mNu }.
(24)
where Di ∈ R1×mNu = {dij}, dij = 1, i = j, dij = 0, i 6= j,. The results are summarized in the following theorem. Theorem 1. The inclusion Ω (P ) ⊆ L(F ) is for output feedback control equivalent to
P Di FC
C T F T DTi
λi
≥ 0,
i = 1, 2, . . . , mNu
(25)
where
λi ∈ 0,
Ui2
θ
.
Proof. To prove that the inclusion Ω (P ) ⊆ L(F ) is equivalent to (25), we use S- procedure in the following way. Rewrite (23) and (24) to the following form p(z ) = z T (t )Pz (t ) − θ ≤ 0 gi (z ) = z T (t )CfT F T DTi Di FCz (t ) − Ui2 ≤ 0.
According to the S-procedure, there exists a positive scalar λi such that gi (z ) − λi p(z ) ≤ 0 or equivalently z (t )T (CfT F T DTi Di FC − λi P )z (t ) − Ui2 + λi θ ≤ 0.
(26)
After some manipulation, (26) can be rewritten in the form C T F T DTi Di FC − λi P 0
0
−Ui2 + λi θ
≤ 0 i = 1, 2, . . . , mNu . (27)
The above inequality for diagonal matrix is equivalent to two inequalities. Using the Schur complement formula for the first one, the inequality (25) is obtained, which proves the theorem. In order to check the value of θi for i-th input, we solve the optimization problem z (t )T Pz (t ) → max subject to constraints (24), whose solution yields
θi =
Ui2 Di FCP −1 C T F T DTi
.
(28)
In the design procedure, it should be verified that when parameter θ decreases, the obtained robust positively invariant regions Ω (P ) are nested to region obtained for θ + , > 0. Assume that we calculate two output feedback gain matrices: F1 for unconstrained case and F2 for constrained one. Obviously, the closed-loop system with the gain matrix F2 gives the dynamic behavior slower than the one obtained for F1 . Consider the output feedback gain matrix F in the form F = γ F1 + (1 − γ )F2 ,
γ ∈ (0, 1).
(29)
For gain matrices Fi , i = 1, 2 we obtain two closed-loop system in the form (13), Aci = Af + Bf Fi Cf , i = 1, 2. Consider the edge between Ac1 and Ac2 , that is Ac = α Ac1 + (1 − α)Ac2 ,
α ∈ h0, 1i.
(30)
The following lemma gives the stability conditions for matrix Ac (30). Lemma 2. Consider the stable closed-loop system matrices Aci , i = 1, 2.
• If there exists a positive definite matrix Pq such that
The above set can be rewritten as L(F ) = {z (t ) ∈ R(nN ) : |Di FCf z (t )| ≤ Ui , i = 1, 2, . . . , mNu }
117
ATci Pq Aci − Pq ≤ 0,
i = 1, 2
(31)
then matrix Ac is quadratically stable. • If there exist two positive definite matrices P1 , P2 such that they satisfy the parameter dependent quadratic stability conditions (see [18,22]), the closed-loop system Ac is parameter-dependent quadratically stable (PDQS). Remarks. • If the closed-loop matrices Aci , i = 1, 2 satisfy (31), the scalar γ in (29) may be changed with any rate without violating the closed-loop stability. • If the closed-loop matrices Aci , i = 1, 2 are PDQS, the scalar γ in (29) has to be constant but may be unknown. • The proposed control algorithm (29) is similar to Soft VariableStructure Control (SVSC) (Adamy and Flemming [23]) but in our case, when |vi | Ui the feedback gain matrix F (29) gives rather quicker dynamic behavior of the closed-loop system (unconstrained case) than when |vi | approaches to Ui . Algorithm for calculation of γ for (29) may be as follows:
γ = min i
Ui − |vi | Ui
.
(32)
118
V. Veselý et al. / ISA Transactions 49 (2010) 114–120
If accidentally some |vi | > Ui , γ = 0. The resulting control design procedure is given by the next steps
0.3 1 0.25 0.2
• Off-line computation stage, compute output feedback gain matrices: F1 for unconstrained case as a solution to (20), where LMI (20) is solved for unknown P and F ; F2 for the constrained case as a solution to (20) and (25). • On-line computation — in each step: compute the actual value of scalar parameter γ ; e.g from (32) (where vi is obtained from (12) for F = F1 ); compute the actual feedback gain matrix from (29) and respective constrained control vector from (12). All on-line computations follows the general MPC scheme; i.e. the first part of computed control vector u(t ) is applied on the real controlled plant and the other part of control vector is used for model prediction.
0.15 0.1 0.05 0 -0.05 -0.1 -0.15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 x 10-5
Fig. 1. Dynamic behavior of controlled system for unconstrained case for u(t ).
0.05 1
5. Examples
0.04 0.03
Two examples are presented to illustrate the qualities of the control design procedure proposed above, namely its ability to cope with robust stability and input constraints without complex computational load. In each example, the results of three simulation experiments are compared for closed-loop with output feedback control:
-0.01
Case 1. Unconstrained case for output feedback gain matrix F1
-0.02
Case 2. Constrained case for output feedback gain matrix F2
-0.03
Case 3. The new proposed control algorithm (29) for output feedback gain matrix F . The input constraint case is studied. In each case maximal value of u(t ) is checked; stability is assessed using spectral radius of closed-loop system matrix. First example serves as a benchmark. The model of double integrator turns to (1) where
Ao =
1 1
0.02 0.01 0
-0.04
0 1
0.4
0.5
0.6
0.7
0.8
0.9 1 x 10-5
0.1 1
Cv = 0
0.04
and uncertain matrices are
0.02
0.01 0.02
0.01 0.03
0.001 . 0
0.3
Fig. 2. Dynamic behavior of controlled system for constrained case for u(t ).
1
B1u =
0.2
0.06
1 , 0
A1u =
0.1
0.08
Bo =
0
For the case when number of uncertainty is p = 1 the number of vertices are K = 2p = 2, the matrices (2) are calculated as follows A1 = Ao − A1u , A2 = Ao + A1u , B1 = Bo − B1u , B2 = Bo + B1u . For the parameters of % = 20000, N = 6, Nu = 6, Q0 = 0.1I , Q1 = 0.5I , Q2 = · · · = Q6 = I , R = I the following results are obtained for unconstrained and constrained cases
• Unconstrained case Closed-loopmaxeig = 0.8495. Maximal value of control variable is about umax = 0.24. • Constrained case with Ui = 0.1, θ = 1000, Closed-loopmaxeig = 0.9437. Maximal value of control variable is about umax = 0.04. Closed-loop step responses for unconstrained and constrained cases are given in Figs. 1 and 2.Closed-loop step responses for the
0 -0.02 -0.04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 x 10-5
Fig. 3. Dynamic behavior of controlled system with the proposed algorithm for u (t ).
case in the proposed algorithm in the paper are given in Fig. 3. Maximal value of control variable is about umax = 0.08 < 0.1. Input constraints conditions were applied only for plant control variable u(t ). Second example has been borrowed from Camacho and Bordons [1] p.147. The model corresponds to the longitudinal motion of a Boeing 747 airplane. The multivariable process is controlled using a predictive controller based on the output model of the aircraft. Two of the usual command outputs that must be controlled are airspeed (that is, velocity with respect to air) and climb rate.
V. Veselý et al. / ISA Transactions 49 (2010) 114–120
10
0.8 1 2
8
0.4
4
0.2
2
0
0
-0.2
-2
-0.4
-4
-0.6
-6
-0.8
-8
1 2
0.6
6
-10
119
0
0.5
1
Fig. 5. Dynamic behavior of constrained controlled system for u(t ).
0
0.5
1
1.5 1
Fig. 4. Dynamic behavior of unconstrained controlled system for u(t ).
.9996 −.0056 Ao = .002 .0001 .0001 −.0615 Bo = −.1133 −.0057
.0383 .9647 −.0097 −.0005 .1002 .0183 .0586 .0029
.0131 .7446 .9543 .0978
-0.5 -1 -1.5
1 1 0
0 −1
0 0
0 7.74
0
A1u
0
B1u
0 0.0005 0 0
0 −0.02 = −0.12 0
0 0.0017 0.0001 0
-2
0
0.5
1
1.5
Fig. 6. Dynamic behavior for proposed control algorithm (29) and (32) for u(t ).
6. Conclusion
and model uncertainty
0 = 0
0
−.0322 .0001 0
Cv =
1 2
0.5
The continuous model has been converted to a discrete time one with a sampling time of 0.1s, the nominal model turns to (1) where
1.5
0 0 0 0
0.12 0.1 −3 10 . 0 0
For the case when the number of uncertainty is p = 1, the number of vertices are K = 2p = 2, and the matrices (2) are calculated as in example 1. Note that nominal model Ao is unstable. Consider N = Nu = 1, % = 20 000 and weighting matrices Q0 = Q1 = 1I , R0 = R1 = I. The following results are obtained:
• Unconstrained case maximal closed-loop nominal model eigenvalue is Closed-loopmaxeig = 0.9983. Maximal value of control variables are about u1max = 9.6, u2max = 6.3. • Constrained case with Ui = 1, θ = 40000 Closed-loopmaxeig = 0.9998 Maximal value of control variables are about u1max = 0.21, u2max = 0.2. Closed-loop nominal model step responses of the above two cases for the input u(t ) are given in the Figs. 4 and 5. Closed-loop step responses for the control algorithm proposed in the paper (29) and (32) are in Fig. 6. Maximal value of control variables are about u1max = 0.75 < 1, u2max = 0.6 < 1. Input constraint conditions were applied only for plant control variable u(t ). Both examples show that, using tuning parameter θ , the demanded input constraints can be reached with high accuracy. The initial guess of θ can be obtained from (28). It can be seen that the proposed control scheme provides reasonable results: the responses in case 3 (Figs. 3 and 6) are quicker than those in case 2 (Figs. 2 and 5), while the computation load has not increased much as compared to case 2.
The paper addresses the problem of designing the output/state feedback robust model predictive controller with input constraints for N and Nu output and control prediction horizon. The main contribution of the presented results is twofold: the obtained robust control algorithm guarantees the closed-loop system quadratic stability and guaranteed cost under input constraints in the whole uncertainty domain. The required on-line computation load is significantly less than in MPC literature (according to the best knowledge of the authors), which opens the possibility to use this control design scheme not only for plants with slow dynamics but also for faster ones. At each sample time, the calculation of the proposed control algorithm reduces to a solution of a simple equation. Finally, two examples illustrate the effectiveness of the proposed method. Acknowledgement The work has been supported by Grant N 1/0544/09 of the Slovak Scientific Grant Agency. References [1] Camacho EF, Bordons C. Model predictive control. Springer-Verlag London Limited; 2004. [2] Maciejovski JM. Predictive Control with Constraints. Prentice Hall; 2002. [3] Rossiter JA. Model based predictive control: A practical approach. Control Series 2003. [4] Zafiriou E, Marchal A. Stability of SISO quadratic dynamic matrix control with hard output constraints. AIChE Journal 1991;37:1550–60. [5] Polak E, Yang TH. Moving horizon control of linear systems with input saturation and plant uncertainty. International Journal of Control 1993;53: 613–38. [6] Zheng Z.Q., Morari M.. Robust stability of constrained model predictive control. In: Proc. ACC. 379–83. 1993. [7] dela Pena D.M., Alamo T., Ramirez T., Camacho E.. Min–max model predictive control as a quadratic program. In: Proc. of 16th IFAC World Congress, CDROM. 2005. [8] Bouzouita B., Bouani F., Ksouri M.. Efficient implementation of multivariable MPC with parametric uncertainties. In: Proc. ECC. TuB12.4, CD-ROM. 2007.
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