Optimal feedback control of non-linear systems

Krikelis, N. J.; Kiriakidis, Kiriakos

doi:10.1080/00207729208949445

Cited by 18 publications

(13 citation statements)

References 2 publications

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“…Figure 5 also gives the cost function comparison which shows that the costs produced by these two method are very close. Figure 6 shows the results when the initial states are [10,10] using the y approximation. The control response is a zoomed plot in Figure 6.…”

Section: A 2-d Benchmark Problem [11]mentioning

confidence: 98%

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A new method for suboptimal control of a class of non-linear systems

Xin

Balakrishnan

2005

Optim. Control Appl. Meth.

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SUMMARYIn this paper, a new non-linear control synthesis technique (y-D approximation) is discussed. This approach achieves suboptimal solutions to a class of non-linear optimal control problems characterized by a quadratic cost function and a plant model that is affine in control. An approximate solution to the Hamilton-Jacobi-Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both semi-global asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large-control-for-large-initial-states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive online computations like the recently popular state dependent Riccati equation (SDRE) technique. Furthermore, it provides a closed-form non-linear feedback controller if finite number of terms are taken in the series expansion. A scalar problem and a 2-D benchmark problem are investigated to demonstrate the effectiveness of this new technique. Both stability and convergence proofs are given.

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Section: A 2-d Benchmark Problem [11]mentioning

confidence: 98%

“…Therefore, as the system order increases, the complexity of determining these coefficients increases dramatically. Zhang et al [9] and Krikelis et al [10] proposed a better way for calculating the coefficients of the series solution given by Garrard. But it can only guarantee the stability around the origin.…”

Section: Introductionmentioning

confidence: 99%

A new method for suboptimal control of a class of non-linear systems

Xin

Balakrishnan

2005

Optim. Control Appl. Meth.

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“…Adding the above two equations and comparing the coefficients give the Riccati matrix equation (10), the Sylvester equation (11) and the adjoint equation…”

Section: Lemma 1 ([14]mentioning

confidence: 99%

“…The proposed algorithm only requires solving the Riccati equation (10) and the Sylvester equation (11) once, and mainly solves an iteration formula of adjoint vector sequence. It takes less computing time and memory space compared with iteration of matrix differential equations.…”

Section: Remarkmentioning

confidence: 99%

“…A second method is the successive Galerkin approximation [10], which reduces the HJB equation to a sequence of linear partial differential equations. Another approach is called the state-dependent Riccati equation [11,12], which requires the solution of the Riccati equation at each given point of the system state. In order to simplify the computation, Aganovic and Gajic [13] proposed a method of successive approximation, where only solutions of a sequence of differential Lyapunov equations are required.…”

Section: Introductionmentioning

confidence: 99%

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Optimal tracking control for large-scale interconnected systems with time-delays

Tang

Sun

2007

Computers & Mathematics with Applications

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This paper considers an infinite-horizon optimal tracking control problem for a class of large-scale interconnected systems with state time-delays. By using the successive approximation approach, two iteration sequences of vector differential equations are constructed. Meanwhile the large-scale interconnected system is decomposed into finite decoupled subsystems. The existence and uniqueness of the optimal solution is proved, as well as the convergence of the solution sequence. By finite iterations of the solution sequence, a suboptimal tracking control law is obtained. A reduced-order reference input observer is designed to make the feedforward term of the optimal tracking control law physically realizable. A numerical example shows that the presented algorithm is effective and easy to implement.

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Reduced‐order suboptimal control design for a class of nonlinear distributed parameter systems using POD and θ–D techniques

Padhi

Xin

Balakrishnan

2007

Optim Control Appl Methods

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A new computational tool is presented in this paper for suboptimal control design of a class of nonlinear distributed parameter systems (DPSs). In this systematic methodology, first proper orthogonal decomposition-based problem-oriented basis functions are designed, which are then used in a Galerkin projection to come up with a low-order lumped parameter approximation. This technique has evolved as a powerful model reduction technique for DPSs. Next, a suboptimal controller is designed using the emerging -D technique for lumped parameter systems. This time domain control solution is then mapped back to the distributed domain using the same basis functions, which essentially leads to a closed form solution for the controller in a state-feedback form. We present this technique for the class of nonlinear DPSs that are affine in control. Numerical results for a benchmark problem as well as for a more challenging representative real-life nonlinear temperature control problem indicate that the proposed method holds promise as a good optimal control design technique for the class of DPSs under consideration. 192 R. PADHI, M. XIN AND S. N. BALAKRISHNAN equations (PDEs). DPSs appear naturally in various application areas such as chemical processes, thermal processes, vibrating structures, fluid flow systems, etc. They inherently have an infinite number of system modes, and hence, are also known as infinite-dimensional systems. Since it is impossible to deal with all the modes, some sort of approximation technique is usually applied for the analysis and synthesis procedures related to DPS. Control of DPS has been studied both from a mathematical and an engineering point of view. An interesting brief historical perspective of the control of such systems can be found in Reference [1].There exist infinite-dimensional operator-theory-based methods for the control of DPSs. While there are many advantages, these operator-theory-based approaches are mainly limited to linear systems [2] and some limited class of problems like spatially invariant systems [3]. Moreover, for implementation purpose, the infinite-dimensional control solution needs to be approximated (e.g. truncating an infinite series, reducing the size of feedback gain matrix, etc.). Such a control design approach is known as 'design then approximate'. Eventhough the resulting controller from this approach retains most of the information of the associated full-order controller, the technique is mainly confined to linear DPS. Besides, it is not completely free from errors because of the approximations involved for its implementation. Another control design approach is 'approximate then design'. Here, the PDEs describing the system dynamics are first approximated to yield a finite-dimensional approximate model. This approximate system is then used for control design. In this approach, it is relatively easy to design controllers using various concepts of finite-dimensional control design. This approach has more engineering relevance, since it can be applied to both linea...

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Optimal feedback control of non-linear systems

Cited by 18 publications

References 2 publications

A new method for suboptimal control of a class of non-linear systems

A new method for suboptimal control of a class of non-linear systems

Optimal tracking control for large-scale interconnected systems with time-delays

Reduced‐order suboptimal control design for a class of nonlinear distributed parameter systems using POD and θ–D techniques

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