Stochastic MPC for Systems with both Multiplicative and Additive Disturbances

Cheng, Qifeng; Cannon, Mark; Kouvaritakis, B.; Evans, Martin

doi:10.3182/20140824-6-za-1003.01255

Cited by 14 publications

(14 citation statements)

References 19 publications

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“…MPC (and stochastic MPC) uses estimated system dynamics to forecast where the system will be in several time steps and then chooses control inputs that will bring that forecast more in line with the target trajectory over that time horizon (Figure 5D; Table 1; Rawlings, 2000; Cheng et al, 2014). Various fast strategies have been developed to solve such problems including precomputed lookup tables (Rauov et al, 2009; Maurovic et al, 2011), suboptimal control strategies (Wang and Boyd, 2010; Bertsekas, 2005a), and fast explicit solutions that may be carried out at each step (Wang and Boyd, 2011b).…”

Section: Implementing Closed-loop Control For Neural Microcircuitsmentioning

confidence: 99%

Closed-Loop and Activity-Guided Optogenetic Control

Grosenick

Marshel

Deisseroth

2015

Neuron

356

322

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Advances in optical manipulation and observation of neural activity have set the stage for widespread implementation of closed-loop and activity-guided optical control of neural circuit dynamics. Closing the loop optogenetically (i.e., basing optogenetic stimulation on simultaneously observed dynamics in a principled way) is a powerful strategy for causal investigation of neural circuitry. In particular, observing and feeding back the effects of circuit interventions on physiologically relevant timescales is valuable for directly testing whether inferred models of dynamics, connectivity, and causation are accurate in vivo. Here we highlight technical and theoretical foundations as well as recent advances and opportunities in this area, and we review in detail the known caveats and limitations of optogenetic experimentation in the context of addressing these challenges with closed-loop optogenetic control in behaving animals.

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Section: Implementing Closed-loop Control For Neural Microcircuitsmentioning

confidence: 99%

Closed-Loop and Activity-Guided Optogenetic Control

Grosenick

Marshel

Deisseroth

2015

Neuron

356

322

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“…Therefore, it is intractable to calculate the probabilistic distribution of e p ( i | k ) given the distribution of w p . To remedy this, following the work of Cheng et al, e p ( i | k ) is further split into 2 parts, which are given for

i \in double-struckN

e_{p} false(i false| k false) = ϵ_{p} false(i false| k false) + ζ_{p} false(i false| k false),

ϵ_{p} false(i + 1 false| k false) = {normalΨ}_{p}^{0} ε_{p} false(i false| k false) + w_{p} false(i false| k false),

ζ_{p} false(i + 1 false| k false) = Ψ_{p} false(i false| k false) ζ_{p} false(i false| k false) + {\overset{Ψ}{true}}_{p} false(i false| k false) z_{p} false(i false| k false) + {\overset{B}{true}}_{p} false(i false| k false) c_{p} false(i false| k false) + {\overset{Ψ}{true}}_{p} false(i false| k false) ϵ_{p} false(i false| k false),

with the initial conditions ϵ p ( k )= 0 and z p ( k )+ ζ p ( k )= x p ( k ). Subsequently, according to the different probabilistic nature, the 2 components of e p (ie, ϵ p and ζ p ) in will be treated separately based on ideas from tube MPC.…”

Section: Probabilistic Constraint Handling Strategymentioning

confidence: 99%

“…To date, methods accounting for both parameter uncertainty and disturbances have only been developed for single systems. () Based on an augmented autonomous state space description in the work of Kouvaritakis et al, a computationally convenient stochastic model predictive control (SMPC) approach was investigated in the work of Cannon et al By using an extension of the concept of invariance, namely, invariance with probability, and confining the augmented state to the invariant set with probability, it is possible to achieve satisfaction of constraints with given probabilities and establish recursive feasibility and closed‐loop stability. Cannon et al only considered a particular type of probabilistic constraints, which take the form of limits on the expected number of samples at which the system output lies outside a desired interval over a given horizon.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the system matrices considered were expressed as affine functions of one scalar uncertain parameter, which limits the applicability of this algorithm. In the work of Cheng et al, the state trajectory was decomposed into 3 parts, which were handled separately to ensure probabilistic constraint satisfaction, and the resulting algorithm provided desirable closed‐loop properties: recursive feasibility and stability.…”

Section: Introductionmentioning

confidence: 99%

“… Inspired by other works,() the uncertain part of the state trajectory is decomposed into 2 parts: one is handled by calculating a sufficient tightening margin offline, whereas the other is handled by using polytopic tubes with facets of fixed orientation and online adjustable scaling. Both local and coupled probabilistic constraints are then reformulated as linear deterministic constraints, further leading to a convex optimization problem, which can be efficiently solved. To ensure coupled probabilistic constraint satisfaction under the distributed update strategy in the work of Trodden and Richards, the constraints derived in the centralized implementation are modified, and the resulting constraints involve only the decision variables of a local subsystem and the prior information collected from coupled subsystems. Through designing terminal sets as a robust positively invariant set and an invariant hypercube with a parameter chosen from a given interval, the DSMPC algorithm is proven to be recursively feasible, and the resulting closed‐loop system is stable. Exchange of data among subsystems is only required after one subsystem optimization at each time step.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Distributed stochastic MPC for systems with parameter uncertainty and disturbances

Xia

Gao

et al. 2018

Intl J Robust & Nonlinear

Self Cite

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Summary A distributed stochastic model predictive control algorithm is proposed for multiple linear subsystems with both parameter uncertainty and stochastic disturbances, which are coupled via probabilistic constraints. To handle the probabilistic constraints, the system dynamics is first decomposed into a nominal part and an uncertain part. The uncertain part is further divided into 2 parts: the first one is constrained to lie in probabilistic tubes that are calculated offline through the use of the probabilistic information on disturbances, whereas the second one is constrained to lie in polytopic tubes whose volumes are optimized online and whose facets' orientations are determined offline. By permitting a single subsystem to optimize at each time step, the probabilistic constraints are then reduced into a set of linear deterministic constraints, and the online optimization problem is transformed into a convex optimization problem that can be performed efficiently. Furthermore, compared to a centralized control scheme, the distributed stochastic model predictive control algorithm only requires message transmissions when a subsystem is optimized, thereby offering greater flexibility in communication. By designing a tailored invariant terminal set for each subsystem, the proposed algorithm can achieve recursive feasibility, which, in turn, ensures closed‐loop stability of the entire system. A numerical example is given to illustrate the efficacy of the algorithm.

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A randomized relaxation method to ensure feasibility in stochastic control of linear systems subject to state and input constraints

2020

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We consider a linear system affected by an additive stochastic disturbance and address the design of a finite horizon control policy that is optimal according to some cost criterion and accounts also for probabilistic constraints on both the input and state variables. The resulting policy can be implemented over a receding horizon according to the model predictive control strategy. Such a possibility, however, is hampered by the fact that a feasibility issue may arise when recomputing the policy. Infeasibility indeed can occur if the disturbance has unbounded support and the state is required to remain in a bounded set. In this paper, we propose a solution to this issue that is based on the introduction of a constraint relaxation that becomes effective only when the original problem turns out to be unfeasible. This is obtained via a cascade of two probabilistically-constrained optimization problems where, in the first one, performance is neglected and the policy is designed to fully recover feasibility or -if this is not possible-to determine the minimum level of relaxation which is needed to recover feasibility; in the second step, such a minimum relaxation level is imposed while optimally (re-)tuning the control policy parameters. Both problems are solved through a computationally tractable scenario-based scheme using a finite number of disturbance realizations and providing an approximate solution that satisfies with high confidence the original probabilistic constraints of the cascade.

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Stochastic MPC for Systems with both Multiplicative and Additive Disturbances

Cited by 14 publications

References 19 publications

Closed-Loop and Activity-Guided Optogenetic Control

Closed-Loop and Activity-Guided Optogenetic Control

Distributed stochastic MPC for systems with parameter uncertainty and disturbances

A randomized relaxation method to ensure feasibility in stochastic control of linear systems subject to state and input constraints

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