Optimal Control Systems by Time-Dependent Coefficients Using CAS Wavelets

Abstract

This paper considers the problem of controlling the solution of an initial boundary-value problem for a wave equation with time-dependent sound speed. The control problem is to determine the optimal sound speed function which damps the vibration of the system by minimizing a given energy performance measure. The minimization of the energy performance measure over sound speed is subjected to the equation of motion of the system with imposed initial and boundary conditions. Using the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of bilinear time-invariant lumped-parameter systems. A wavelet-based method for evaluating the modal optimal control and trajectory of the bilinear system is proposed. The method employs finite CAS wavelets to approximate modal control and state variables. Numerical examples are presented to demonstrate the effectiveness of the method in reducing the energy of the system.

1. Introduction

Dynamic stability related to parametric resonance is a very important factor in structural dynamics. For example, instability caused by parametric resonance is believed to be the reason for the famous Tacoma bridge collapse in 1940 [1]. A suitable control of the coefficients may provide an effective protection against this phenomenon.

Control in the coefficients is known to be a very effective method in structures governed by elliptic equations [1]. However, not much information is known about the effect produced by control in coefficient for hyperbolic equations [2, 3]. In this paper, we study a control problem for a structure dynamic system governed by a hyperbolic equation where the control is a time dependent coefficient.

The model considered in this work is motivated by recent developments in the area of smart materials [4]. The properties of these materials can be changed by applying external fields, such as electrical, magnetic, or temperature; this is referred to as a phase transformation.

A structure made with such a material is considered where control consists of eliminating a transient disturbance in the structure by varying the material properties in the response to the deformation. The modal dynamic of a structure is governed by a scalar wave equation, where the control variable is the sound speed in the medium. The basic bilinear optimal control problem becomes the minimization of the energy function of the system in a given period of time with a minimum sound speed. Using modal expansion, the optimal control of the distributed parameter system is reduced to the optimal control of a bilinear time-varying lumped parameter system. The parameterization approach is used to approximate the state-variable and each component of the control variable using finite-term wavelets with unknown coefficients. Therefore, the quadratic problem is transformed into a mathematical programming problem with the objective of minimizing the unknown coefficients to give suboptimal solution of the problem. A necessary condition for the optimality of the unknown coefficients is derived as a system of linear algebraic equations for which the solution is used to obtain the optimal control sound speed and optimal state function.

The bilinear system is a kind of nonlinear system where some related problems such as optimal control are much more difficult to solve than those of linear systems. In literature, many authors [5–9] have tried various methods to overcome the difficulties of solving bilinear systems. In this paper, the focus will be on obtaining the optimal state solution of a wave equation governed by a bilinear system using CAS wavelets taking advantage of some needed properties of this type of wavelets [10, 11]. Compared to conventual method such as Fourier series or finite elements, CAS wavelets with their local properties enable arbitrary functions (even with discontinuity) to be approximated more efficient. To demonstrate the effectiveness of the proposed approach, numerical results will show confirm that the proposed method significantly minimizes the energy of the system.

2. Optimal Control Problem Setting

Let Ω_𝑥 be an open, bounded, and simply connected subset of 𝑛-dimensional Euclidean space ℝ^𝑛. Let Ω_𝑡 denote a given time interval (0,𝑡_𝑓) with finite terminal time 𝑡_𝑓. Consider the wave equation, defined on 𝑄=Ω_𝑥×Ω_𝑡: 𝑢_𝑡𝑡=𝑎(𝑡)Δ𝑢, (2.1) where Δ is the Laplacian operator, and 𝑢(𝑥,𝑡) is the disturbance of position 𝑥 and time 𝑡. The wave speed 𝑎(𝑡) is assumed to be a function of time. For simplicity, let 𝑢 satisfy the boundary and initial conditions: 𝑢(𝑥,𝑡)=0,Δ𝑢=0,∀𝑥∈𝜕Ω_𝑥,𝑢(𝑥,0)=𝑤₀(𝑥),𝑢_𝑡(𝑥,0)=𝑤₁(𝑥),𝑥∈Ω_𝑥, (2.2) where 𝑤₀(𝑥)∈𝐻²(Ω_𝑥𝜕)={ℎ(𝑥)∶^𝑖ℎ𝜕𝑥^𝑖∈𝐿²(Ω_𝑥),𝑖=1,2}, (2.3) and 𝑤₁(𝑥)∈𝐿²(Ω_𝑥).

Let the admissible control set be 𝐴_ad={𝑎(𝑡)∶𝑎(𝑡)∈𝐿²(Ω_𝑡)}. (2.4) Associated with the wave equation (2.1) is the modified energy 𝐽[𝑎(𝑡)] at terminal time 𝑡_𝑓: 𝐽[𝑎(𝑡)]=𝜇_1Ω𝑥𝑢²(𝑥,𝑡_𝑓)𝑑𝑥+𝜇_2Ω𝑥𝑢²_𝑡(𝑥,𝑡_𝑓)𝑑𝑥+𝜇_3Ω𝑡𝑎²(𝑡)𝑑𝑡, (2.5) where 𝜇₁, 𝜇₂, and 𝜇₃ are weighing constants satisfying the condition 𝜇₁+𝜇₂>0 and 𝜇₃>0. The last term on the right-hand side of (2.5) is a penalty term on control energy.

The optimal control problem is stated as follows: determine the optimal control function 𝑎^∗(𝑡)∈𝐴_adsuch that𝐽[𝑎^∗(𝑡)]=min_{𝑎(𝑡)∈𝐴ad}𝐽[𝑎(𝑡)] (2.6)subject to (2.1) and (2.2).

3. Control Problem in Modal Space

We pose the problem at hand as a control problem for an finite system of ordinary differential equations by using modal space expansion. Let 𝑢(𝑥,𝑡)=^𝑁_𝑛=1𝑧_𝑛(𝑡)𝜑_𝑛(𝑥), (3.1) where 𝜑_𝑛(𝑥) are normalized eigenfunctions associated with eigenvalues 𝑤²_𝑛. This implies that 𝜑_𝑛(𝑥) satisfies the eignvalue-problem Δ𝜑_𝑛(𝑥)+𝑤²_𝑛𝜑_𝑛(𝑥)=0,𝑥∈Ω_𝑥,𝜑_𝑛(𝑥)=0,𝑥∈𝜕Ω_𝑥,Δ𝜑_𝑛(𝑥)=0,𝑥∈𝜕Ω_𝑥. (3.2) It can be shown that the set 𝜑_𝑛(𝑥) forms an orthonormal set, and hence 𝑧_𝑛(𝑡) satisfies 𝑑²𝑑𝑡²𝑧_𝑛(𝑡)+𝑎(𝑡)𝑤²_𝑛𝑧_𝑛(𝑡)=0,𝑛=1,2,…,𝑁 (3.3) with initial conditions 𝑧_𝑛(0)=𝑧_0𝑛,𝑑𝑧𝑑𝑡^𝑛(0)=𝑧_1𝑛,𝑛=1,…,𝑁. (3.4) In view of the expansion (3.1), the performance index becomes𝐽_𝑁[𝑎(𝑡)]=^𝑁_𝑛=1[𝜇₁𝑧²_𝑛(𝑡_𝑓)+𝜇₂(𝑑𝑧𝑑𝑡^𝑛(𝑡_𝑓))²]+𝜇₃^𝑡𝑓₀𝑎²(𝑡)𝑑𝑡. (3.5) The optimal control problem (2.6) is now modified as follows: determine 𝑎^∗(𝑡)∈𝐴_adsuch that𝐽_𝑁[𝑎^∗(𝑡)]=min_{𝑎(𝑡)∈𝐴ad}𝐽_𝑁[𝑎(𝑡)] (3.6)subject to (3.3) and (3.4).

4. Properties of the CAS Wavelets

4.1. CAS Wavelets

Wavelets have been used by many scientists and engineers to solve several problems in areas such as signal and image processing, control problems, and stochastic problems. Wavelets are mathematical functions that are constructed using dilation and translation of a single function called the mother wavelet denoted by 𝜓(𝑡) and must satisfy certain requirements. If the dilation parameter is 𝑎 and translation parameter is 𝑏, then we have the following family of wavelets: 𝜓_𝑎,𝑏(𝑡)=|𝑎|^−1/2𝜓(𝑡−𝑏𝑎),with𝑎,𝑏∈𝑅,𝑎≠0. (4.1) Restricting 𝑎 and 𝑏 to discrete values, such as 𝑎=𝑎₀^−𝑘, 𝑏=𝑛𝑏₀𝑎₀^−𝑘, 𝑎₀>1, 𝑏₀>0 and 𝑛 and 𝑘 are positive integers, gives 𝜓_𝑘,𝑛(𝑡)=|𝑎₀|^𝑘/2𝜓(𝑎^𝑘₀𝑡−𝑛𝑏₀), (4.2) where 𝜓_𝑘,𝑛(𝑡) form a basis for 𝐿²(𝑅). If 𝑎₀=2 and 𝑏₀=1, then it is clear that the set {𝜓_𝑘,𝑛(𝑡)} forms an orthonormal basis for 𝐿²(𝑅).

The CAS wavelets employed in this paper are defined as 𝜓_𝑛,𝑚2(𝑡)=^𝑘/2CAS_𝑚(2^𝑘𝑛𝑡−𝑛),if2^𝑘≤𝑡<𝑛+12^𝑘,0,otherwise, (4.3) where CAS_𝑚(𝑡)=cos(2𝑚𝜋𝑡)+sin(2𝑚𝜋𝑡). (4.4) The set of CAS wavelets forms an orthonormal basis for 𝐿²([0,1]). This implies that any function 𝑓(𝑡) defined over [0,1] can be expanded as 𝑓(𝑡)=^∞_{𝑛=0𝑚∈𝑍}𝑑_𝑛,𝑚𝜓_𝑛,𝑚≃(𝑡)_{2𝑘−1𝑀𝑛=0𝑚=−𝑀}𝑑_𝑛,𝑚𝜓_𝑛,𝑚(𝑡)=𝐷^𝑇Ψ(𝑡), (4.5) where 𝑑_𝑛,𝑚=(𝑓(𝑡),𝜓_𝑛,𝑚(𝑡))=¹₀𝑓(𝑡)𝜓_𝑛,𝑚(𝑡)𝑑𝑡, (4.6) and 𝐷 and Ψ(𝑡) are 2^𝑘(2𝑀+1)×1 vectors given by 𝐷=[𝑑_0,−𝑀,𝑑_{0,(−𝑀+1),}…,𝑑_0,𝑀,𝑑_1,−𝑀,…,𝑑_1,𝑀,…,𝑑₍₂^𝑘_{−1),−𝑀},…,𝑑₍₂^𝑘_−1),𝑀]^𝑇,Ψ(𝑡)=[Ψ_0,−𝑀,Ψ_{0,(−𝑀+1)},…,Ψ_0,𝑀,Ψ_1,−𝑀,…,Ψ_1,𝑀,…,Ψ₍₂^𝑘_{−1),−𝑀},…,Ψ^𝑇_{(2𝑘−1),𝑀}]. (4.7)

4.2. Operational Matrices of Integration

The integration of the function Ψ(𝑡) in (4.5) is given by ^𝑡₀Ψ(𝑠)𝑑𝑠=𝑃Ψ(𝑡), (4.8) where 𝑃 is an 2^𝑘(2𝑀+1)×2^𝑘(2𝑀+1) matrix, called the operational matrix, and is given by [12] 1𝑃=2^𝑘+1𝐹𝑆𝐹𝐹⋯𝐹𝑂𝑆𝐹⋯𝐹⋮𝑂⋱⋱⋮𝑂𝑂⋯𝑂𝑆 (4.9) in which 𝑂 is a zero matrix and 𝐹 and 𝑆 are (2𝑀+1)×(2𝑀+1) matrices given by ,1𝐹=0⋯⋯⋯0⋮⋱⋮⋱0⋯2⋯0⋮⋮⋮⋱⋱0⋯⋯⋯0𝑆=00⋯0−1𝑀𝜋0⋯01𝑀𝜋00⋯0−1(𝑀−1)𝜋0⋯01(𝑀−1)𝜋⋯⋯⋯⋯⋯⋯⋯⋯⋯00⋯0−𝜋1𝜋1⋯00𝜋1𝜋⋯1𝜋11𝜋⋯1𝜋1𝜋100⋯𝜋1𝜋010⋯00⋯⋯⋯⋯⋯⋯⋯⋯⋯1(𝑀−1)𝜋⋯01(𝑀−1)𝜋0⋯001𝑀𝜋0⋯0.𝑀𝜋0⋯00 (4.10) The integration of the product of two CAS function vectors is given by ¹₀Ψ(𝑡)Ψ(𝑡)^𝑇𝑑𝑡=𝐼. (4.11) The product operational matrix of the CAS wavelet is given by Ψ(𝑡)Ψ(𝑡)^𝑇𝐶≃𝐶Ψ(𝑡), (4.12) where the matrix 𝐶 is given in (4.7) and 𝐶 is an 2^𝑘(2𝑀+1)×2^𝑘(2𝑀+1) given by [11] 𝐶𝐶=¹00𝐶₂, (4.13) where 𝐶_𝑖, 𝑖=1,2 are (2𝑀+1)×(2𝑀+1) given by 𝐶_𝑖=𝑐_𝑖,0𝑐_𝑖,−10𝑐_𝑖,−1𝑐_𝑖,0𝑐_𝑖,10𝑐_𝑖,1𝑐_𝑖,0for𝑖=1,2. (4.14)

5. CAS-Wavelets-Based Approach

To redefine the wavelet functions over the interval [0,𝑡_𝑓], we let 𝑡=𝑡_𝑓𝜏. Then (3.3), (3.4), and (3.5), respectively, become 𝑑²𝑑𝜏²𝑧_𝑖(𝜏)+𝑤²_𝑖𝑡²_𝑓𝑎(𝜏)𝑧_𝑖𝑧(𝜏)=0,for0<𝜏<1,𝑖=1,2,…,𝑁,_𝑖(0)=𝑧_𝑖0,𝑑𝑧𝑑𝜏^𝑖(0)=𝑡_𝑓𝑧_𝑖1𝐽,𝑖=1,…,𝑁,_𝑁(𝑎(𝜏))=^𝑁_𝑖=1[𝜇₁𝑧²_𝑖(𝑡_𝑓)+𝑡²_𝑓𝜇₂𝑑𝑧𝑑𝜏^2𝑖(𝑡_𝑓)]+𝑡_𝑓𝜇₃¹₀𝑎²(𝜏)𝑑𝜏. (5.1)(5.2)

Using the expansion in (4.5) gives 𝑑²𝑑𝜏²𝑧_𝑖(𝜏)=𝐵^→Ψ_𝑖^𝑇→(𝜏)=Ψ^→𝑇(𝜏)𝐵^→_𝑖,𝑎(𝜏)=𝐶^→Ψ^𝑇→(𝜏)=Ψ^→𝑇(𝜏)𝐶^→,𝑧_𝑖(0)=Φ^→Ψ^𝑇_𝑖^→(𝜏)=Ψ^→𝑇(𝜏)Φ^→_𝑖,𝑑𝑧𝑑𝜏^𝑖(0)=^→ϝΨ^𝑇_𝑖^→(𝜏)=Ψ^→𝑇(𝜏)ϝ^→_𝑖, (5.3) where 𝐵^→_𝑖, 𝐶^→, Φ^→_𝑖, ^→ϝ_𝑖, and Ψ^→(𝜏) are 2^𝑘(2𝑀+1)×1 vectors defined as in (4.7). Furthermore, 𝑑𝑧𝑑𝜏^𝑖(𝜏)=^𝜏₀𝑑²𝑑𝑠²𝑧_𝑖𝑑(𝑠)𝑑𝑠+𝑧𝑑𝜏^𝑖=(0)_𝜏0→𝐵Ψ_𝑖^𝑇→(𝑠)𝑑𝑠+^→ϝ^𝑇_𝑖^→=Ψ(𝜏)_→𝐵_𝑖^𝑇𝑃Ψ^→(𝜏)+^→ϝ^𝑇_𝑖^→𝑧Ψ(𝜏),_𝑖(𝜏)=^𝜏₀𝑑𝑧𝑑𝜏^𝑖(𝑠)𝑑𝑠+𝑧_𝑖=(0)_𝜏0(𝐵^→_𝑖^𝑇𝑃Ψ^→(𝑠)+^→ϝΨ^𝑇_𝑖^→(𝑠))𝑑𝑠+Φ^→𝑇_𝑖^→=Ψ(𝜏)_→𝐵_𝑖^𝑇𝑃Ψ^2→(𝜏)+^→ϝ𝑃^𝑇_𝑖^→Ψ(𝜏)+Φ^→𝑇_𝑖^→=Ψ(𝜏)_→Ψ^𝑇(𝜏)(𝑃²)𝐵^𝑇→_𝑖+Ψ^→𝑇(𝜏)𝑃ϝ^𝑇→_𝑖+Ψ^→𝑇(𝜏)Φ^→_𝑖. (5.4)(5.5) Substituting (5.5) in (5.1) yields𝐵Ψ^→_𝑖^𝑇→(𝜏)+𝑤²_𝑖𝑡𝐶²_𝑓^→Ψ^𝑇→(𝜏)[Ψ^𝑇(𝜏)(^→𝑃²)𝐵^𝑇→_𝑖+Ψ^→𝑇(𝜏)𝑃ϝ^𝑇→_𝑖+Ψ^→𝑇(𝜏)Φ^→_𝑖]=0 (5.6) and hence 𝐵Ψ^→_𝑖^𝑇→(𝜏)+𝑤²_𝑖𝑡𝐶²_𝑓^→Ψ^𝑇→(𝜏)Ψ^𝑇(𝜏)(𝑃²)𝐵^𝑇→_𝑖+𝑤²_𝑖𝑡𝐶²_𝑓^→Ψ^𝑇→(𝜏)Ψ^→𝑇(𝜏)𝑃ϝ^𝑇→_𝑖+𝑤²_𝑖𝑡𝐶^→2_𝑓^→Ψ^𝑇→(𝜏)Ψ^𝑇(𝜏)Φ^→_𝑖=0. (5.7)

Using (4.12) leads to 𝐵Ψ^→_𝑖^𝑇→(𝜏)+𝑤²_𝑖𝑡Ψ²_𝑓^→𝑇(𝜏)𝐶^𝑇(𝑃²)𝐵^𝑇→_𝑖+𝑤²_𝑖𝑡Ψ²_𝑓^→𝑇(𝜏)𝐶^𝑇𝑃ϝ^𝑇→_𝑖+𝑤²_𝑖𝑡Ψ²_𝑓^→𝑇(𝜏)𝐶Φ^𝑇→_𝑖=0. (5.8)

Multiplying (5.8) by Ψ^→(𝜏), integrating, and using (4.11) give 𝐵^→_𝑖+𝑤²_𝑖𝑡²_𝑓𝐶^𝑇(𝑃²)𝐵^𝑇→_𝑖+𝑤²_𝑖𝑡²_𝑓𝐶^𝑇𝑃ϝ^𝑇→_𝑖+𝑤²_𝑖𝑡²_𝑓𝐶Φ^𝑇→_𝑖=0 (5.9) or 𝐵^→_𝑖=−𝐺⁻¹(𝑤²_𝑖𝑡²_𝑓𝐶^𝑇𝑃ϝ^𝑇→_𝑖+𝑤²_𝑖𝑡²_𝑓𝐶Φ^𝑇→_𝑖) (5.10) provided that 𝐺=(𝐼+𝑤²_𝑖𝑡²_𝑓𝐶^𝑇(𝑃²)^𝑇)⁻¹ (5.11) exists. Substituting equations (5.4) and (5.5) into equation (3.5) convert the performance index 𝐽_𝑁[𝑎(𝑡)] into a function of 𝐶 and hence to optimize 𝐽_𝑁[𝑎(𝑡)], we solve 𝜕𝐽_𝑁𝜕𝑐_𝑖=0. (5.12)

6. Numerical Example

Consider the wave equation 𝑑²𝑑𝜏²𝑧(𝜏)+𝑤²𝑡²_𝑓𝑎(𝜏)𝑧(𝜏)=0,for0<𝜏<1 (6.1) with initial conditions 𝑑𝑧(0)=1,𝑑𝜏𝑧(0)=0. (6.2)

For the sake of illustration, the following parameters were assumed: 𝑤₁=𝜋,𝑡_𝑓=1,𝜇₁=𝜇₂=𝜇₃=1,Ω_𝑥=(0,1),𝑀=1,𝑘=1(6waveletexpansions). (6.3) The performance index was computed for the optimal control 𝑎^∗(𝑡) and compared with the performance index for the controllers 𝑎(𝑡)=𝑡, 𝑎(𝑡)=𝑡², 𝑎(𝑡)=sin𝑡, and 𝑎(𝑡)=cos𝑡. The results are summarized in Table 1.

Table 1 

Comparison between uncontrolled and controlled performance indices.

It is observed that the proposed control is effective in significantly reducing the performance index of the problem.

7. Conclusion

A control for a wave equation where the control is a time dependent coefficient is considered. A modal space technique simplifies the optimal control of a distributed parameter system into the optimal control of a bilinear time-invariant lumped-parameter system. A Galerkin CAS wavelet-based method was developed to solve this bilinear optimal control problem. The main aspect of the proposed approach resides in converting the optimization problem into a mathematical programming problem where the necessary conditions of optimality are derived as a system of algebraic equations . A test example, which includes a variable coefficient and one-dimensional hyperbolic equation, demonstrates the capability of the proposed Galerkin-Wavelet approach for solving optimal control problems governed by bilinear systems. Moreover, the numerical simulations show that the optimal control procedure led to a substantial damping in the bilinear system energy.

This method may be extended to treat a more general setting where the coefficients are 𝑥 and 𝑡 dependent. That is, the wave speed function is a controllable function of the form 𝑎(𝑥,𝑡) [13].