blocking된 모델예측제어의 강건성

이광순, 문혜진, 김상훈

서강대학교 화학공학과

Robustness of blocked Model Predictive Control

Kwang Soon Lee, Hye Jin Moon, Sang Hoon Kim

Department of Chemical Engineering, Sogang University

Introduction

Through the development of last fifteen years, Model Predictive Control(MPC) is now regarded as a standard control paradigm for multivariable process control. In MPC, to find the optimal control moves which satisfy the quadratic objective with constraints, many MPC algorithms use the quadratic programming(QP). In general, QP requires heavy computational efforts. So virtually all commercial packages employ blocking techniques or other to reduce computational efforts. Here, blocking refers to any approximation method for input movement over the control horizon in order to reduce the dimensionality of input calculation.

As implied by name, MPC needs an model. Given the realities of process identification, modelling error is unavoidable. Therefore, one of the major problems with MPC is robustness analysis.

Based on the above considerations, the focus of this paper is placed on the analysis of robust stability of MPC which has blocking techniques.

MPC mathmatical formulation

For a state space model of process rearranged to have an integrated input, the optimal state estimator is the optimal filter of the following form.

Process model

[eqn: Y_k = M_p Y_k-1 + S^u DELTA U_k-1 +S^d DELTA d_k-1]	(1)

[eqn: hat y_k = N_p Y_k + v_k`]	(2)

Prediction

[eqn: Y_k|k-1 = M_p Y_k-1|k-1 + S^u DELTA U_k-1 +S^d DELTA d_k-1]	(3)

[eqn: y_k|k-1 = N_p Y_k|k-1]	(4)

Correction

[eqn: Y_k|k = Y_k|k-1 + K_f `(hat y_k - y_k|k-1 )~``~`]	(5)

In the above, [eqn: S^u] is the dynamic matrix and [eqn: hat y_k] is the measurement of output at time step [eqn: k=k`].

The solution of MPC can be obtained by calculating the input moves [eqn: DELTA U_k] which minimizes an objective such as:

[eqn: MATRIX { {min }#{DELTAU_k } } ~ ||` LAMBDA _y`` `( R_k+1|k - Y_k+1|k`` )``||_2^2 ``+`` ||` LAMBDA_u`` DELTA U_k`` ||_2^2]	(6)

subject to

[eqn: Y_k+1|k = M_p Y_k|k + S^u DELTA U_k +S^d DELTA d_k`]

[eqn: C_u DELTA U_k >= C_k]

In a typical implementation, QP problem given by equations (6) is solved on-line and the control moves are implemented in a receding horizon fashion, i.e., only the first move of [eqn: DELTA U_k`] is implemented and the optimization is repeated for the next sampling time.

Blocking of MPC

The mathematical formulation of MPC was presented. Blocked MPC can be obtained by replacing input movements [eqn: DELTA U_k`] with blocked input movements [eqn: F_u` DELTA U_k^*`]. Here, the input movements [eqn: DELTA U_k`] are approximated by [eqn: b`] basis vectors.

[eqn: DELTA U_k `=`` DELTA u _1^* `f_1 + DELTA u_2^* `f_2 + cdots + DELTA u_m^* `f_m # CONG`` DELTA u _1^* `f_1 + DELTA u_2^* `f_2 + cdots + DELTA u_b^* `f_b ~` # CONG ``F_u DELTA U _k^*~~~~~~~~~~~~~~~~ (`b`<m`)]	(7)

Then, the objective (6) can be rewritten in terms of the blocked vectors as follows:

[eqn: MATRIX { {min }#{DELTAU_k^* } } ~ ||` LAMBDA _y`` `epsilon_k+1|k^* ``||_2^2 ``+`` ||` LAMBDA_u`` F_u`` DELTA U_k^*`` ||_2^2]	(8)

subject to

[eqn: epsilon_k+1|k^*``= ` R_k+1|k `-`M_p `Y_k|k `-`S^d `DELTA d_k` -` S^u `F_u `DELTA U_k^*]	(9)

The unconstrained solution of the above objective (8) is

[eqn: DELTA U_k^* = [F_u^T S^u^T LAMBDA_y^T LAMBDA_y S^u F_u + F_u^T LAMBDA_u^T LAMBDA_u F_u ]^-1 F_u^T S^u^T LAMBDA_y^T LAMBDA_y [R_k+1|k - M_p Y_k|k - S^d DELTA d_k ]]

········(10)

[eqn: DELTA U_k = F_u DELTA U_k^* = K_mpc ``epsilon_k+1|k`]	(11)

Here,

[eqn: K_mpc = F_u `[F_u^T S^u^T LAMBDA_y^T LAMBDA_y S^u F_u + F_u^T LAMBDA_u^T LAMBDA_u F_u ]^-1 F_u^T S^u^T LAMBDA_y^T LAMBDA_y`]	(12)

[eqn: epsilon_k+1|k = [R_k+1|k - M_p Y_k|k - S^d DELTA d_k ]]	(13)

Discrete Cosine Transform

In this work, we will use the Discrete Cosine Transform (DCT) as a blocking technique. DCT is a special case of Discrete Fourier Transform (DFT), composed of only cosine terms by half-range expansion. DCT closely approximates the Karhunen - Loeve transform (KLT) especially for highly correlated signals. As is well-known, KLT is the optimal transform which can minimize the truncation error among other transforms with the same number of basis functions. One of the drawbacks of KLT is that it is signal-dependent. On the other hand, DCT is independent of signals while it retains the properties of KLT. Since the output predictions and input moves are inevitably highly correlated by the process model, DCT can be considered a proper candidate for blocking of MPC. The fast algorithm of generating the DCT matrix defined over [0, 1, ... , N-1] is as follows:

[eqn: PHI (k,n) = PHI_k (n) = ( 1over c_k ) cos {(2n+1)k pi}over{2N} , ~~0<= n,k <= N-1]	(14)

[eqn: c_k = CASES{ {root N, ~~~~~~~~~k=0} #{root N/2 , ~~~~~~~ k`` != 0}}]	(15)

In the above, [eqn: omega_k = ( 2pi over 2N ) k`] has the physical meaning of the associated frequency.

Numerical examples

In this subsection, numerical examples are provided to demonstrate the performance of the blocking technique using DCT. Two different cases are considered; one case where the model is exactly known and the other where high frequency model uncertainty is there. The control performance by blocking is almost preserved in the first case and in the second case robustness gained by the proposed blocking is demonstrated.

First, in case of no model error, the plant and output disturbance models are derived from the following transfer function matrices with sampling interval [eqn: h=1].

♠plant	[eqn: G_p = PMATRIX{ {4.05e^-27 over 50s+1 }& {1.77e^-28 over 60s+1 }# { 5.39e^-18 over 50s+1}& {5.72e^-14 over 60s+1 } }]	(16)

♠output disturbance	[eqn: G_d = PMATRIX{ {1.2e^-27 over 45s+1 }& {1.44e^-27 over 40s+1 }# { 1.52e^-15 over 25s+1}& {1.83e^-15 over 20s+1 } }]	(17)

♠constraints	[eqn: -1<= U_1,k , U_2,k <= 1 ~~ and ~~ DELTA U_k <= 0.2]	(18)

Second, in case of model error, the nominal model is assumed to have model mismatch in the diagonal elements of the transfer function matrix as given in the following equations.

Fig 1. MPC (p=32, m=32)

Fig 2. DCT (c=4, b=4)

♠plant     [eqn: G_p = PMATRIX {{2 OVER {(10s+1)(10s^2 +s+1)}} &{2 over 30s+1}#{2 over 20s+1}&{2 OVER {(10s+1)(5s^2 +s+1)}}}]	(19)

♠nominal model	[eqn: G_m = pmatrix {{2 over 10s+1 }&{2 over 30s+1}#{2 over 20s+1}&{2 over 10s+1}}]	(20)

♠output disturbances	[eqn: G_d = PMATRIX{ {1e^-7 over 45s+1 }& {1e^-7 over 40s+1 }# { 1e^-5 over 25s+1}& {1e^-5 over 20s+1 } }]	(21)

♠constraints	[eqn: -1<= U_1,k , U_2,k <= 1 ~~ and ~~ DELTA U_k <= 0.2]	(22)

Fig . MPC (p=32, m=32)

Fig . DCT (c=2, b=2)

Conclusion

In this work, our main objective is to analysis the robustness of MPC using blocking technique. We used the DCT as a blocking technique because crisp frequency resolution of discrete cosine functions make it possible to cut off the undesired frequency range where significant model error is expected. As a result, when there is no model error, the performance was almost preserved. And when there is uncertainty, the robust stability was obtained by the proposed blocking technique. Several simulations and theoretical analyses were performed to show the above result.

References

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