Control of nonlinear systems with explicit-MPC-like controllers

Pejcic, Ivan; Korda, Milan; Jones, Colin N.

doi:10.1109/cdc.2017.8264394

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…where J * N →N (x) = x x, and J * k→N (•) is the optimal cost-togo of the step k, and (19a) denotes the Euler discretization of ( 18), and T s is the sampling time, and X k+1→N is the (N −k−1)-step stabilizable set. The quadratic cost function in (19) selects controllers that result in fast convergence by not penalizing u(x), making it competitive to CPA designs of this paper. To compute approximations of {X k→N } N −1 k=0 and {J * k→N (•)} N −1 k=0 , a uniform grid for the state-space with step size δx (1) = δx (2) is assumed.…”

Section: A Controllersmentioning

confidence: 99%

“…Thus at each grid point, a feasibility problem with linear constraints is formulated for each k ∈ Z N −1 k=0 . Then, (19) is solved backwards at the grid points inside the stabilizable sets given J * N →N (x) and using interpolation to evaluate optimal cost-to-goes.…”

Section: A Controllersmentioning

confidence: 99%

“…For instance, model predictive control (MPC) implies existence of a Lyapunov function by appropriate choices of terminal 'ingredients' [16] like the terminal set, terminal cost, and terminal stabilizing controller. Finding an explicit Lyapunov function and the controller can only be done afterwards via taxing explicit nonlinear MPC [17]- [19]. Also, Lyapunov-based methods rely on finding a Lyapunov-like function such as control Lyapunov function (CLF) or control barrier function (CBF) first [9], [12], [13], [20]- [22], and even with one at hand, the controller is usually realized online using quadratic programming (QP), which is not desirable when computing resources are limited.…”

Section: Introductionmentioning

confidence: 99%

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Systematic, Lyapunov-Based, Safe and Stabilizing Controller Synthesis for Constrained Nonlinear Systems

Reza¹,

Bridgeman²

2022

Preprint

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A controller synthesis method for state-and input-constrained nonlinear systems is presented that seeks continuous piecewise affine (CPA) Lyapunov-like functions and controllers simultaneously. Non-convex optimization problems are formulated on triangulated subsets of the admissible states that can be refined to meet primary control objectives, such as stability and safety, alongside secondary performance objectives. A multi-stage design is also given that enlarges the region of attraction (ROA) sequentially while allowing exclusive performance for each stage. A clear boundary for an invariant subset of closed-loop system's ROA is obtained from the resulting Lipschitz Lyapunov function. For control-affine nonlinear systems, the non-convex problem is formulated as a series of conservative, but well-posed, semi-definite programs. These decrease the cost function iteratively until the design objectives are met. Since the resulting CPA Lyapunov-like functions are also Lipschitz control (or barrier) Lyapunov functions, they can be used in online quadratic programming to find minimum-norm control inputs. Numerical examples are provided to demonstrate the effectiveness of the method.

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Section: A Controllersmentioning

confidence: 99%

Section: A Controllersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Systematic, Lyapunov-Based, Safe and Stabilizing Controller Synthesis for Constrained Nonlinear Systems

Reza¹,

Bridgeman²

2022

Preprint

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“…In this case, the corresponding Lipschitz CLF is used to formulate a minimumnorm controller by QP. In seeking both the controller and the Lyapunov function offline, this method is similar to [16], but it is not limited to polynomial systems. Like [10], the method depends on refining elements in a subset of the state space, but it avoids solving the highly nonlinear optimization.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Controller and Lyapunov Function Design for Constrained Nonlinear Systems

Reza¹,

Bridgeman²

2021

Preprint

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This paper presents a method to stabilize state and input constrained nonlinear systems using an offline optimization on variable triangulations of the set of admissible states. For control-affine systems, by choosing a continuous piecewise affine (CPA) controller structure, the non-convex optimization is formulated as iterative semi-definite program (SDP), which can be solved efficiently using available software. The method has very general assumptions on the system's dynamics and constraints. Unlike similar existing methods, it avoids finding terminal invariant sets, solving non-convex optimizations, and does not rely on knowing a control Lyapunov function (CLF), as it finds a CPA Lyapunov function explicitly. The method enforces a desired upper-bound on the decay rate of the state norm and finds the exact region of attraction. Thus, it can be also viewed as a systematic approach for finding Lipschitz CLFs in state and input constrained control-affine systems. Using the CLF, a minimum norm controller is also formulated by quadratic programming for online application.

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“…For linear cases, since there is a structure to be exploited that results in less computational load, MPC is especially preferred. However, depending on the availability of computation power and the importance of the application, MPC is also deployed in many nonlinear systems [15,16]. Using MPC-based methods, it is also possible to address some specific controllability and observability problems for LTI or nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Comparative Study of an EKF-Based Parameter Estimation and a Nonlinear Optimization-Based Estimation on PMSM System Identification

Sel

Coskun

et al. 2021

Energies

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In this study, two different parameter estimation algorithms are studied and compared. Iterated EKF and a nonlinear optimization algorithm based on on-line search methods are implemented to estimate parameters of a given permanent magnet synchronous motor whose dynamics are assumed to be known and nonlinear. In addition to parameters, initial conditions of the dynamical system are also considered to be unknown, and that comprises one of the differences of those two algorithms. The implementation of those algorithms for the problem and adaptations of the methods are detailed for some other variations of the problem that are reported in the literature. As for the computational aspect of the study, a convexity study is conducted to obtain the spherical neighborhood of the unknown terms around their correct values in the space. To obtain such a range is important to determine convexity properties of the optimization problem given in the estimation problem. In this study, an EKF-based parameter estimation algorithm and an optimization-based method are designed for a given nonlinear dynamical system. The design steps are detailed, and the efficacies and shortcomings of both algorithms are discussed regarding the numerical simulations.

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Control of nonlinear systems with explicit-MPC-like controllers

Cited by 8 publications

References 11 publications

Systematic, Lyapunov-Based, Safe and Stabilizing Controller Synthesis for Constrained Nonlinear Systems

Systematic, Lyapunov-Based, Safe and Stabilizing Controller Synthesis for Constrained Nonlinear Systems

Simultaneous Controller and Lyapunov Function Design for Constrained Nonlinear Systems

Comparative Study of an EKF-Based Parameter Estimation and a Nonlinear Optimization-Based Estimation on PMSM System Identification

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