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Beyond Persistent Excitation: Online Experiment Design for Data-Driven Modeling and Control
Abstract
This letter presents a new experiment design method for data-driven modeling and control. The idea is to select inputs
online
(using past input/output data), leading to desirable rank properties of data Hankel matrices. In comparison to the classical persistency of excitation condition, this online approach requires less data samples and is even shown to be completely sample efficient.
... e HΛ − Λ) = 0 can be verified through standard computations. We can therefore conclude that (14) is ...
... As such, Lemma 6 asserts that det(H) ̸ = 0. Then the proposed active learning strategy guarantees that a full rank data matrix H will be obtained after the phase of formation. In particular, when we consider a noise-free controllable system and replace (31) with an optimization problem searching for a positive solution, the proposed strategy degenerates to the online experiment design method introduced in [14]. Namely, any u that makes J F (H, x, u) ̸ = 0 is a feasible solution for the experiment design problem in [14] and vice versa. ...
... In particular, when we consider a noise-free controllable system and replace (31) with an optimization problem searching for a positive solution, the proposed strategy degenerates to the online experiment design method introduced in [14]. Namely, any u that makes J F (H, x, u) ̸ = 0 is a feasible solution for the experiment design problem in [14] and vice versa. ...
An important question in data-driven control is how to obtain an informative dataset. In this work, we consider the problem of effective data acquisition of an unknown linear system with bounded disturbance for both open-loop and closed-loop stages. The learning objective is to minimize the volume of the set of admissible systems. First, a performance measure based on historical data and the input sequence is introduced to characterize the upper bound of the volume of the set of admissible systems. On the basis of this performance measure, an open-loop active learning strategy is proposed to minimize the volume by actively designing inputs during the open-loop stage. For the closed-loop stage, an closed-loop active learning strategy is designed to select and learn from informative closed-loop data. The efficiency of the proposed closed-loop active learning strategy is proved by showing that the unselected data cannot benefit the learning performance. Furthermore, an adaptive predictive controller is designed in accordance with the proposed data acquisition approach. The recursive feasibility and the stability of the controller are proved by analyzing the effect of the closed-loop active learning strategy. Finally, numerical examples and comparisons illustrate the effectiveness of the proposed data acquisition strategy.
... We provide an example of this in Section 8. The online experiment design method of [30] keeps the depth of the data Hankel matrix fixed during the operation of the algorithm. In contrast to [30], our approach adapts the depth of the Hankel matrix during operation of the algorithm. ...
... The online experiment design method of [30] keeps the depth of the data Hankel matrix fixed during the operation of the algorithm. In contrast to [30], our approach adapts the depth of the Hankel matrix during operation of the algorithm. The rationale is that the "correct" depth of the Hankel matrix for the necessary and sufficient conditions of [5] is a priori unknown. ...
... The rationale is that the "correct" depth of the Hankel matrix for the necessary and sufficient conditions of [5] is a priori unknown. We show in Section 8 that our method outperforms [30] in some situations, depending on the given L and N. In other cases, our results prove that the approach of [30] leads to the shortest experiments for linear system identification. ...
This paper is concerned with the following problem: given an upper bound of the state-space dimension and lag of a linear time-invariant system, design a sequence of inputs so that the system dynamics can be recovered from the resulting input-output data. As our main result we propose a new online experiment design method, meaning that the selection of the inputs is iterative and guided by data samples collected in the past. We show that this approach leads to the shortest possible experiments for linear system identification. In terms of sample complexity, the proposed method outperforms offline methods based on persistency of excitation as well as existing online experiment design methods.
... Generalizations to uncontrollable systems are in [15][16][17] and extensions to continuoustime systems in [18][19][20][21]. Quantitative/robust variations are explored in [22,23], frequency domain formulations in [24,25], and online experiment design in [26]. Furthermore, the fundamental lemma has been generalized beyond linear systems to include various other model classes: descriptor systems [27], flat nonlinear systems [28], linear parameter-varying systems [29], and stochastic ones [30]. ...
... ; by applying the rank-nullity theorem to (26), we see that ...
... To prove (b), note first that J k (x) has full row rank whenever k ∈ [0, i] due to (47). For the rest, observe that (26) implies rank H k = rank Φ k and rank G k = rank Ψ k whenever J k (x) has full row rank. From the definitions, we have rank Φ k = (k+1)m+rank Ω k and rank Ψ k = (k + 1)m + rank Ω k−1 ; the claim is proved. ...
We state necessary and sufficient conditions to uniquely identify (modulo state isomorphism) a linear time-invariant minimal input-state-output system from finite input-output data and upper- and lower bounds on lag and state space dimension.
... In general, there exist only few results on the design of suitable inputs that result in satisfaction of the required PE conditions, in particular for nonlinear systems. In [7], an online method was proposed to design inputs that result in the desired rank conditions of [1] on input-state or input-output Hankel matrices for linear systems. However, the resulting input is not universal in the sense that it is tailored specifically to the system on which the experiment is performed. ...
... Definition 1). In contrast, the online procedure from [7] directly designs an input such that the resulting input-output data matrix in (2) has rank mL + n and requires N = (m + 1)L + n − 1 samples to this end, implying that it is sample efficient. The input (3), although not sample efficient, guarantees PE independently of the considered system (contrary to the online procedure of [7]). ...
... In contrast, the online procedure from [7] directly designs an input such that the resulting input-output data matrix in (2) has rank mL + n and requires N = (m + 1)L + n − 1 samples to this end, implying that it is sample efficient. The input (3), although not sample efficient, guarantees PE independently of the considered system (contrary to the online procedure of [7]). Finally, an appealing feature of the input (3) is its high sparsity. ...
In the context of data-driven control, persistence of excitation (PE) of an input sequence is defined in terms of a rank condition on the Hankel matrix of the input data. For nonlinear systems, recent results employed rank conditions involving collected input and state/output data, for which no guidelines are available on how to satisfy them a priori. In this paper, we first show that a set of discrete impulses is guaranteed to be persistently exciting for any controllable LTI system. Based on this result, for certain classes of nonlinear systems, we guarantee persistence of excitation of a sequence of basis functions a priori, by design of the physical input only. Finally, for nonlinear systems which are locally reachable at the origin, we show that there exist sparse input sequences that guarantee collective PE of sequences of basis functions.
... The persistency of excitation condition imposes a lower bound on the required number of data samples. As shown for the first time in [22], for a single controllable system, one can improve the sample efficiency by generating an online experiment of shorter length that still enables the parametrization of all system trajectories. Moreover, [23] constructed the shortest possible experiment for linear system identification, based on the necessary and sufficient conditions in [24]. ...
... In this sense, persistently exciting inputs are universal as they work for the whole class of controllable systems. This is in contrast to online experiment design methods in [22], [23] that are tailored to a specific controllable system, because the online input design is guided by the outputs of the data-generating system. The existing literature lacks a full characterization of universal inputs. ...
In this letter, we provide new insight into Willems et al.’s fundamental lemma by studying the concept of universal inputs. An input is called universal if, when applied to any controllable system, it leads to input-output data that parametrizes all finite trajectories of the system. By the fundamental lemma, inputs that are persistently exciting of sufficiently high order are universal. The main contribution of this work is to prove the converse. Therefore, universality and persistency of excitation are equivalent.
... The idea has also been utilized in a receding horizon fashion for model predictive control (MPC) [11], [12] and distributed MPC [13] when data are only locally available to nodes of a network. Considering data richness and noise, [12], [14] study the online implementation of sample efficient data-driven control with noisy system measurements. Aligned with this body of work, the informativity approach to data-driven control [15] considers measurements that do not contain enough information to obtain a unique system. ...
... By definition, we have Q " pI nT d´E d qP d . Lemma 4.2: (Modified closed-loop data-based representation): Let Assumption 1 hold and consider the data matrices Z and Q defined in (14). The system (1) under the controller (10) 1 Let the permutation matrix T F correspond to the permutation tuple π, which is a reordering of the set t1, 2,¨¨¨, nT d u. ...
This paper studies the data-driven control of unknown linear-threshold network dynamics to stabilize the state to a reference value. We consider two types of controllers: (i) a state feedback controller with feed-forward reference input and (ii) an augmented feedback controller with error integration. The first controller features a simpler structure and is easier to design, while the second offers improved performance in the presence of system parameter changes and disturbances. Our design strategy employs state-input datasets to construct data-based representations of the closed-loop dynamics. Since these representations involve linear threshold functions, we rewrite them as switched linear systems, and formulate the design problem as that of finding a common controller for all the resulting modes. This gives rise to a set of linear matrix inequalities (LMIs) whose solutions corresponds to the controller gain matrices. We analyze the computational complexity of solving the LMIs and propose a simplified, sufficient set of conditions that scales linearly with the system state. Simulations on two case studies involving regulation of firing rate dynamics in rodent brains and of arousal level dynamics in humans demonstrate the effectiveness of the controller designs.
... The persistency of excitation condition imposes a lower bound on the required number of data samples. As shown for the first time in [22], for a single controllable system, one can improve the sample efficiency by generating an online experiment of shorter length that still enables the parametrization of all system trajectories. Moreover, [23] constructed the shortest possible experiment for linear system identification, based on the necessary and sufficient conditions in [24]. ...
... In this sense, persistently exciting inputs are universal as they work for the whole class of controllable systems. This is in contrast to online experiment design methods in [22], [23] that are tailored to a specific controllable system, because the online input design is guided by the outputs of the data-generating system. The existing literature lacks a full characterization of universal inputs. ...
In this letter, we provide new insight into Willems et al.'s fundamental lemma by studying the concept of universal inputs. An input is called universal if, when applied to any controllable system, it leads to input-output data that parametrizes all finite trajectories of the system. By the fundamental lemma, inputs that are persistently exciting of sufficiently high order are universal. The main contribution of this work is to prove the converse. Therefore, universality and persistency of excitation are equivalent.
... Indeed, in existing work on the data informativity framework, controllers are constructed offline in one shot, using a batch of data. Online experiment design procedures have been studied in [37]: however, such procedures only aim at generating a batch of informative data and do not incorporate a control objective, as is customary in adaptive control. This motivates us to further investigate the relation between MRAC and data informativity. ...
... Remark 1: (Design of the input). Following [37], the input u r (t) in Lemma 4 can be designed as follows: first, find ξ ∈ R n , η ∈ R m \ {0} such that (27) holds; then, design u r (t) such that (28) holds. ...
The goal of model reference adaptive control (MRAC) is to ensure that the trajectories of an unknown dynamical system track those of a given reference model. This is done by means of a feedback controller that adaptively changes its gains using data collected online from the closed-loop system. One of the approaches to solve the MRAC problem is to impose conditions on the data that guarantee convergence of the gains to a solution of the so-called matching equations. In the literature, various extensions of the concept of persistent excitation have been proposed in an effort to weaken the conditions on the data required for this convergence.Despite these efforts, it is not well-understood what are the weakest possible data requirements ensuring convergence of MRAC. In this paper, we propose a new framework to study the MRAC problem, using the concept of data informativity. Our main contribution is to provide \textit{necessary and sufficient} conditions for the convergence of the adaptive gains to a solution of the matching equations. These necessary and sufficient conditions can be readily checked online as new data are generated by the closed-loop system. Our results reveal that existing excitation conditions impose stronger requirements on the collected data than required. Notably, the necessary and sufficient conditions provided in this paper are weaker than those for unique system identification.
... , u on (T on − 1) such that the corresponding online data (U on − , X on ) are informative for mode detection. To obtain such inputs, we adapt the experiment design method of [31]. When each of the modes (Â i ,B i ) of the system are controllable, [31, Theorem 1] gives a construction for inputs u on (0), . . . ...
... Now, either x(t) ∈ im X on − , in which case, take u(t) equal to 0. In the other case, [31,Theorem 1] shows that there exist η, ξ such that η = 0 and ...
This paper considers the stabilization of unknown switched linear systems using data. Instead of a full system model, we have access to a finite number of trajectories of each of the different modes prior to the online operation of the system. On the basis of informative enough measurements, formally characterized in terms of linear matrix inequalities, we design an online switched controller that alternates between a mode detection phase and a stabilization phase. Since the specific currently-active mode is unknown, the controller employs the most recent online measurements to determine it by implementing computationally efficient tests that check compatibility with the set of systems consistent with the pre-collected measurements. The stabilization phase applies the stabilizing feedback gain corresponding to the identified active mode and monitors the evolution of the associated Lyapunov function to detect switches. When a switch is detected, the controller returns to the mode-detection phase. Under average dwell- and activation-time assumptions on the switching signal, we show that the proposed controller guarantees an input-to-state-like stability property of the closed-loop switched system. Various simulations illustrate our results.
... Specific results can be given when (u, d) follows suitable distributions, for instance (6) holds with probability 1 already for T = n + m when (u, d) is normal i.i.d. [34]. ...
... We will now exploit (34) to show that, under (26), (η1P , K, G) is feasible for (11). To this end, rewrite (34) compactly as Θ + I 0 where Θ is as in (25). ...
The linear quadratic regulator (LQR) problem is a cornerstone of automatic control, and it has been widely studied in the data-driven setting. The various data-driven approaches can be classified as indirect (i.e., based on an identified model) versus direct or as robust (i.e., taking uncertainty into account) versus certainty-equivalence. Here we show how to bridge these different formulations and propose a novel, direct, and regularized formulation. We start from indirect certainty-equivalence LQR, i.e., least-square identification of state-space matrices followed by a nominal model-based design, formalized as a bi-level program. We show how to transform this problem into a single-level, regularized, and direct data-driven control formulation, where the regularizer accounts for the least-square data fitting criterion. For this novel formulation we carry out a robustness and performance analysis in presence of noisy data. In a numerical case study we compare regularizers promoting either robustness or certainty-equivalence, and we demonstrate the remarkable performance when blending both of them.
... , u on (T d −1) such that the corresponding online data (U on − , X on ) are informative for mode identification. To obtain such inputs, we adopt the experiment design method of [21], which constructs inputs u on (0), . . . , u on (n + m − 1) such that the corresponding Σ(U on − , X on ) is a singleton set. ...
... The strategy updates the online data U on − , X on − and a list P match containing all the modes that are compatible with the online data for each time instance. To bound the destabilizing effect of the mode detection phase, the algorithm modifies the experiment design method in [21] (Steps 1 to Steps 6) to add a parameter u max > 0 bounding the magnitude of the detection input. The existence of η ∈ R n satisfying the conditions in Step 2 is guaranteed when each mode is controllable. ...
This work studies data-driven switched controller design for discrete-time switched linear systems. Instead of having access to the full system dynamics, an initialization phase is performed, during which noiseless measurements of the state and the input are collected for each mode. Under certain conditions on these measurements, we develop a stabilizing switched controller for the switched system. To be precise, the controller switches between identifying the active mode of the system and applying a predetermined stabilizing feedback. We prove that if the system switches according to certain specifications, this controller stabilizes the closed-loop system. Simulations on a network example illustrate our approach.
... [4]). By optimality, we have that J(α * ) ≤ J(ᾱ), or (29) = c c ...
... This condition can be checked after collecting the input-output data, but can, in general, not be enforced a priori by suitable design of the input sequence as it is the case for DT-LTI systems (cf. [29]). Studying conditions on {u k } N −1 k=0 that result in persistency of excitation of {Ψ k (u, Ξ)} N −1 k=0 for certain choices of basis functions and/or classes of unknown nonlinearities will be a topic of future research. ...
We present an extension of Willems' Fundamental Lemma to the class of multi-input multi-output discrete-time feedback linearizable nonlinear systems. We study the effect of approximating the unknown nonlinearities with a choice of basis functions that depend only on input and output data, then provide error bounds on the results of the data-based simulation and output matching control problems. Furthermore, we use this data-based approximation of the trajectories of the nonlinear system to design a multi-step robust data-based nonlinear predictive control scheme. We show that this control scheme is recursively feasible and renders the closed-loop system practically exponentially stable. Finally, we illustrate our results on a model of a fully-actuated double inverted pendulum.
... The DD-DC computes the control signal directly from historical pairings by solving Eq. 4 for u t , thereby obviating the need for an explicit representation of the dynamical system (A, b, and C) and the latent state,x, in the controller. Initially formulated for ideal, noise-free, linear dynamics in offline settings with extensive datasets (24), DD-DC has recently been expanded to accommodate noisy observations, nonlinear dynamics, online applications, and limited datasets (26,(43)(44)(45)(46) making it a potent model of computation in biological neurons. In the following sections, we explore the implications of this hypothesis and demonstrate its alignment with existing experimental evidence and original analysis. ...
In the quest to model neuronal function amid gaps in physiological data, a promising strategy is to develop a normative theory that interprets neuronal physiology as optimizing a computational objective. This study extends current normative models, which primarily optimize prediction, by conceptualizing neurons as optimal feedback controllers. We posit that neurons, especially those beyond early sensory areas, steer their environment toward a specific desired state through their output. This environment comprises both synaptically interlinked neurons and external motor sensory feedback loops, enabling neurons to evaluate the effectiveness of their control via synaptic feedback. To model neurons as biologically feasible controllers which implicitly identify loop dynamics, infer latent states, and optimize control we utilize the contemporary direct data-driven control (DD-DC) framework. Our DD-DC neuron model explains various neurophysiological phenomena: the shift from potentiation to depression in spike-timing-dependent plasticity with its asymmetry, the duration and adaptive nature of feedforward and feedback neuronal filters, the imprecision in spike generation under constant stimulation, and the characteristic operational variability and noise in the brain. Our model presents a significant departure from the traditional, feedforward, instant-response McCulloch–Pitts–Rosenblatt neuron, offering a modern, biologically informed fundamental unit for constructing neural networks.
... In order to verify whether (50) is satisfied without access to disturbance data, one can evaluate the left-hand side of (51). However, note that the PE condition in Assumption 3 is only sufficient and not necessary for (50); we refer to [43] for a further discussion on this topic. ...
We present a stochastic constrained output-feedback data-driven predictive control scheme for linear time-invariant systems subject to bounded additive disturbances. The approach uses data-driven predictors based on an extension of Willems' fundamental lemma and requires only a single persistently exciting input-output data trajectory. Compared to current state-of-the-art approaches, we do not rely on availability of exact disturbance data. Instead, we leverage a novel parameterization of the unknown disturbance data considering consistency with the measured data and the system class. This allows for deterministic approximation of the chance constraints in a sampling-based fashion. A robust constraint on the first predicted step enables recursive feasibility, closed-loop constraint satisfaction, and robust asymptotic stability in expectation under standard assumptions. A numerical example demonstrates the efficiency of the proposed control scheme.
... For the sake of sampling efficiency, we note that one might obtain u [0,T−1] via closed-loop experiments. 48,49 For the online optimization phase, we first assume OCP (20) is feasible with the measured initial condition (25) at time instant k = 0. Then, we update polynomial bases and initial conditions for each time instant k ∈ N + , in Steps 3-8. ...
Data‐driven predictive control based on the fundamental lemma by Willems et al. is frequently considered for deterministic LTI systems subject to measurement noise. However, little has been done on data‐driven stochastic control. In this paper, we propose a data‐driven stochastic predictive control scheme for LTI systems subject to possibly unbounded additive process disturbances. Based on a stochastic extension of the fundamental lemma and leveraging polynomial chaos expansions, we construct a data‐driven surrogate optimal control problem (OCP). Moreover, combined with an online selection strategy of the initial condition of the OCP, we provide sufficient conditions for recursive feasibility and for stability of the proposed data‐driven predictive control scheme. Finally, two numerical examples illustrate the efficacy and closed‐loop properties of the proposed scheme for process disturbances governed by different distributions.
... Secondly, we stress that we explicitly use a qLPV representation of a nonlinear process and, thus, in many case, this requirement often boils down to having only u as persistently exciting, since ρ is a function of the inputs and outputs and thus {ρ(k) ⊗ u(k)} N −1 k=0 in general also becomes persistently exciting. For the case of redundant scheduling variables, the persistently excitation condition over uρ can be softened using adequate experimental setups, as discussed in the recent letter [21]. We note that this topic is out of the scope of this work and, thus, we henceforth consider that all u and uρ signals are persistently exciting of sufficient order. ...
Recent literature has shown how linear time-invariant (LTI) systems can be represented by trajectories features, that is relying on a single input-output (IO) data dictionary to span all possible system trajectories, as long as the input is persistently exciting. The so-called behavioural framework is a promising alternative for controller synthesis without the necessity of system identification. In this paper, we benefit from differential inclusion in order to adapt previous results to the case quasi-Linear Parameter Varying (qLPV) embeddings, which are use to represent nonlinear dynamical systems along suitable IO coordinates. Accordingly, we propose a set of data-driven analysis tools for a wide class of nonlinear systems, which enable nonlinear data-driven simulation and predictions. Furthermore, a parameter-dependent dissipativity analysis verification setup is also presented, which serves to assess stability of the system within a bounded operation region. The major requirement is that there should exist a scheduling function which maps the nonlinear outputs into a finite number of scheduling variables, and this function should be analytically known. The effectiveness of the proposed tools is tested in practice and shown to provide accurate descriptions of the nonlinear dynamics by the means of a linear representation structure. For such, we consider a high-fidelity nonlinear simulator of a rotational pendulum benchmark simulator and an electro-mechanical positioning experimental validation test-bench. We also show that, even if the scheduling function is erroneously selected, the proposed framework is still able to offer a trustworthy representation of the output dynamics.
... Further investigation into excitation signal design is outside our scope, but see[43],[44] for recent theoretical results. ...
To address the control challenges associated with the increasing share of inverter-connected renewable energy resources, this paper proposes a direct data-driven approach for fast frequency control in the bulk power system. The proposed control scheme partitions the power system into control areas, and leverages local dispatchable inverter-based resources to rapidly mitigate local power imbalances upon events. The controller design is based directly on historical measurement sequences, and does not require identification of a parametric power system model. Theoretical results are provided to support the approach. Simulation studies on a nonlinear three-area test system demonstrate that the controller provides fast and localized frequency control under several types of contingencies.
... The fundamental lemma is thus an experiment design result, that provides a guide for choosing the inputs of the experiment in order to generate informative data (for system identification). It can be shown that the persistency of excitation condition can be replaced by an online design of the inputs [40], which uses less data samples. An important topic for future work, however, is to develop experiment design methods corresponding to the various problems studied in this paper, especially those for noisy data. ...
The goal of this paper is to provide a tutorial on the so-called informativity framework for direct data-driven analysis and control. This framework achieves certified data-based analysis and control by assessing system properties and determining controllers for sets of systems unfalsified by the data. We will first introduce the informativity approach at an abstract level. Thereafter, we will report case studies where we highlight the strength of the framework in the context of various problems involving both noiseless and noisy data. In particular, we will treat controllability and stabilizability, and stabilization, linear quadratic regulation, and tracking and regulation using exact input-state measurements. Thereafter, we will treat dissipativity analysis, stabilization, and H_inf control using noisy input-state data. Finally, we will study dynamic measurement feedback stabilization using noisy input-output data. We will provide several examples to illustrate the approach. In addition, we will highlight the main tools underlying the framework, such as quadratic matrix inequalities in robust control and quadratic difference forms in behavioral systems theory.
... In [7], signal design for non-iterative direct datadriven techniques was considered for data-driven control, where the amplitude spectrum was designed to minimize the degradation caused by noise. An online method using past input-output data was presented in [8], where it was shown to be sample efficient. In [9], two signal design criteria involving sensitivity functions were proposed and analyzed, for the identification of systems with uncertainties. ...
This paper considers the shaping of amplitude spectra of perturbation signals for the identification of a thermostat system. The current approach in control engineering practice utilizes flat spectrum signals, which may not result in the highest possible accuracy. This research aims to investigate the effectiveness of optimal signals with amplitude spectra designed using two state-of-the-art software approaches, namely the model-based optimal signal excitation 2 (MOOSE2) design and the optimal excitation (optexcit) design, in improving estimation accuracy. Such a comparison on a real system is currently lacking. In particular, there exists a research gap on how the combined choice of signal and model structure affects performance measures. In this research, two model structures are used, which are the autoregressive with exogenous input (ARX) and the output error (OE) model structures. Four performance measures are compared, namely the determinant of the covariance matrix of the parameter estimates and the minimum error, mean error and maximum error in the frequency response. Results show that the optimal signals are effective in reducing the determinant of the covariance matrix and the maximum error in the frequency response for the thermostat system, when applied in combination with the ARX model structure. The flat spectrum signal remains very useful as a general broadband perturbation signal as it provides a good overall fit of the frequency response. The findings from this work highlight the benefits of applying optimal signals especially if the identification results are to be used for control, since these signals improve key performance measures which have direct implications on controller design.
... For the sake of sampling efficiency, we note that one might obtain u [0,T −1] via closed-loop experiments. [48], [49] Algorithm 1 Stochastic data-driven predictive control algorithm with OCP (20) Input: T , N , L w , q ← 0,Ṽ 0 ← +∞, k ← 0 Data collection and pre-processing: If (u, w) [0,T −1] is persistently of exciting of order less than N + n x + 1, go to Step 1, else go to Step 5 5: Determine P , K, G, Γ by (23) and (24) (20) is feasible and V m k ≤J k then q ← 0 5: else backup initial condition: 6: For the online optimization phase, we first assume OCP (20) is feasible with the measured initial condition (25) at time instant k = 0. Then, we update polynomial bases and initial conditions for each time instant k ∈ N + , in Step 3-8. ...
Data-driven predictive control based on the fundamental lemma by Willems et al. is frequently considered for deterministic LTI systems subject to measurement noise. However, little has been done on data-driven stochastic control. In this paper, we propose a data-driven stochastic predictive control scheme for LTI systems subject to possibly unbounded additive process disturbances. Based on a stochastic extension of the fundamental lemma and leveraging polynomial chaos expansions, we construct a data-driven surrogate Optimal Control Problem (OCP). Moreover, combined with an online selection strategy of the initial condition of the OCP, we provide sufficient conditions for recursive feasibility and for stability of the proposed data-driven predictive control scheme. Finally, two numerical examples illustrate the efficacy and closed-loop properties of the proposed scheme for process disturbances governed by different distributions.
... Since the system undergoes no switching during the interval [−T, −1], and the sequence {u(−T ), . . . , u(−1)} is persistently exciting 2 For related work on selecting a suitable input sequence so as to preserve persistence of excitation, see [40]. ...
Motivated by the goal of learning controllers for complex systems whose dynamics change over time, we consider the problem of designing control laws for systems that switch among a finite set of unknown discrete-time linear subsystems under unknown switching signals. To this end, we propose a method that uses data to directly design a control mechanism without any explicit identification step. Our approach is online, meaning that the data are collected over time while the system is evolving in closed-loop, and are directly used to iteratively update the controller. A major benefit of the proposed online implementation is therefore the ability of the controller to automatically adjust to changes in the operating mode of the system. We show that the proposed control mechanism guarantees stability of the closed-loop switched linear system provided that the switching is slow enough. Effectiveness of the proposed design technique is illustrated for two aerospace applications.
... (i) two large step load changes of different sizes in one area, the first being small enough to compensate with local IBR 3 Further investigation into the design of practical input signals for data collection is deferred to future work, but see [30], [31] for recent theoretical results. Fig. 3: Cyber-physical system illustrating frequency control approach [3]. ...
We develop and test a data-driven and area-based fast frequency control scheme, which rapidly redispatches inverter-based resources to compensate for local power imbalances within the bulk power system. The approach requires no explicit system model information, relying only on historical measurement sequences for the computation of control actions. Our technical approach fuses developments in low-gain estimator design and data-driven control to provide a model-free and practical solution for fast frequency control. Theoretical results and extensive simulation scenarios on a three area system are provided to support the approach.
... The fundamental lemma from Willems et al. (2005) is a powerful result that enables the characterization of the subspace of all possible trajectories of a linear timeinvariant (LTI) system using raw time series data sorted into a Hankel matrix. The result has inspired many methods for data-driven analysis and control; see (Markovsky and Rapisarda, 2008;van Waarde et al., 2020;van Waarde, 2021;Coulson et al., 2019;De Persis and Tesi, 2019;Berberich et al., 2020), and the survey by Markovsky and Dörfler (2021) and references therein. ...
The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and control. Under the hood, the fundamental lemma works by applying a persistently exciting input to the system. This ensures that the Hankel matrix of resulting input/output data has the "right" rank, meaning that its columns span the entire subspace of trajectories. However, such binary rank conditions are known to be fragile in the sense that a small additive noise could already cause the Hankel matrix to have full rank. Therefore, in this extended abstract we present a robust version of the fundamental lemma. The idea behind the approach is to guarantee certain lower bounds on the singular values of the data Hankel matrix, rather than mere rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of a deeper input Hankel matrix is sufficiently large. This inspires a new quantitative and robust notion of persistency of excitation. The relevance of the result for data-driven control will also be highlighted through comparing the predictive control performance for varying degrees of persistently exciting data.
... The problem of designing control laws directly using measured data is becoming increasingly prominent in the systems and control literature [1,8,9,11,12,[16][17][18][20][21][22][23]26,27,32,39,44,49]. This "one shot" paradigm has several potential advantages over the two-step procedure of system identification combined with model-based controller synthesis. ...
This paper studies several problems related to quadratic matrix inequalities (QMI's), i.e., inequalities in the semidefinite ordering involving quadratic functions of matrix variables. In particular, we provide conditions under which the solution set of a QMI is nonempty, convex, bounded, or has nonempty interior. We also provide a parameterization of the solution set of a given QMI. In addition, we state projection results, that characterize a subset of "structured" solutions to a QMI. Thereafter, we derive matrix versions of the classical S-lemma and Finsler's lemma, that provide conditions under which all solutions to one QMI also satisfy another QMI. The results will be compared to related work in the robust control literature, such as the full block S-procedure and Petersen's lemma, and it is demonstrated how existing results can be obtained from the results of this paper as special cases. Finally, we show how the various results for QMI's can be applied to the problem of data-driven stabilization. This problem involves finding a stabilizing feedback controller for an unknown dynamical system on the basis of a finite set of noisy data samples. We provide general necessary and sufficient conditions for data-based quadratic stabilization. In addition, we demonstrate how to reduce the computational complexity of data-based stabilization by leveraging QMI projection results. This involves separating the computation of the Lyapunov function and the controller, and also leads to explicit formulas for data-guided feedback gains.
... It *This work was supported by the research environment NewLEADS-New Directions in Learning Dynamical Systems, contract 2016-06079; and Wallenberg AI, Autonomous Systems and Software Program (WASP), funded by Knut and Alice Wallenberg Foundation. 1 was introduced and proven in the behavioral setting in [8], and also reformulated and proven in terms of statespace systems in [9]. The lemma was successfully applied to data-driven simulation and control in [10], data-driven predictive control [11], linear quadratic regulator [12], experiment design [13], [14], as well as other extensions. ...
Although the data-driven control design techniques have gotten attention recently, few results address the tracking problem using data-based control formulations. Considering this, this paper presents a data-driven control method to drive the output of a system to its desired constant reference without using any model knowledge. To this end, first, this paper shows how to take the integral action into account for the data-driven control design for the reference tracking problem. Second, a closed-loop representation of the data-driven tracking controller with integral action is derived. Then, based on the representation, linear matrix inequality (LMI) conditions are derived for gain selection. The effectiveness of the proposed method is demonstrated using the speed control of a DC motor. Thanks to the integral action property, it is shown that the tracking error remains zero under constant disturbance affecting the system.
Recent studies highlight the importance of persistently exciting condition in single signal sequence for model identification and data-driven control methodologies. However, maintaining prolonged excitation in control signals introduces significant challenges, as continuous excitation can reduce the lifetime of mechanical devices. In this paper, we introduce three informativity conditions for various types of multi-signal data, each augmented by weight factors. We explore the interrelations between these conditions and their rank properties in linear time-invariant systems. Furthermore, we introduce open-loop experimental design methods tailored to each of the three conditions, which can synthesize the required excitation conditions either offline or online, even in the presence of limited information within each signal segment. We demonstrate the effectiveness of these informativity conditions in least-squares identification. Additionally, all three conditions can extend Willems' fundamental lemma and are utilized to assess the properties of the system. Illustrative examples confirm that these conditions yield satisfactory outcomes in both least-squares identification and the construction of data-driven controllers.
This paper considers the stabilization of unknown switched linear systems using data. Instead of a full system model, we have access to a finite number of trajectories of each of the different modes prior to the online operation of the system. On the basis of informative enough measurements, we design an online switched controller that alternates between a mode detection phase and a stabilization phase. Since the currently-active mode is unknown, the controller employs online measurements to determine it by implementing computationally efficient tests that check compatibility with the set of systems consistent with the pre-collected measurements. The stabilization phase applies a stabilizing feedback gain corresponding to the identified active mode and monitors the evolution of the associated Lyapunov function to detect switches. When a switch is detected, the controller returns to the mode-detection phase. Under average dwell- and activation-time assumptions on the switching signal, we show that the proposed controller guarantees a practical stability property of the closed-loop switched system. Various simulations illustrate our results.
In the literature, a recent debate has been brought up regarding how linear time-invariant systems can be represented by trajectories features. That is, how a single input–output (IO) data dictionary can be exploited to span all possible system trajectories, as long as the input is persistently exciting. Indeed, the so-called behavioural framework is a promising alternative for controller synthesis without the necessity of system identification procedures. In this paper, we provide an overview of the available results. In particular, we focus on how quasi-Linear Parameter Varying (qLPV) embeddings, in the data-driven context, can be used to represent nonlinear dynamical systems along suitable IO coordinates. We debate the topics of nonlinear data-driven simulation and predictions, as proposed in recent works. The effectiveness of the surveyed tools is tested in practice and shown to provide accurate descriptions of the nonlinear dynamics by the means of a linear representation structure. For such, we consider a high-fidelity nonlinear simulator of a rotational pendulum benchmark simulator and an electro-mechanical positioning experimental validation test-bench. We also debate that, even if the qLPV scheduling function is erroneously selected, the framework is still able to offer a reasonably trustworthy representation of the system.
This work introduces the Data-Enabled Predictive iteRative Control (DeePRC) algorithm, a direct data-driven approach for iterative LTI systems. The DeePRC learns from previous iterations to improve its performance and achieves the optimal cost. By utilizing a tube-based variation of the DeePRC scheme, we propose a two-stage approach that enables safe active exploration using a left-kernel-based input disturbance design. This method generates informative trajectories to enrich the historical data, which extends the maximum achievable prediction horizon and leads to faster iteration convergence. In addition, we present an end-to-end formulation of the two-stage approach, integrating the disturbance design procedure into the planning phase. We showcase the effectiveness of the proposed algorithms on a numerical experiment.
To address the control challenges associated with the increasing share of inverter-connected renewable energy resources, this paper proposes a direct data-driven approach for fast frequency control in the bulk power system. The proposed control scheme partitions the power system into control areas, and leverages local dispatchable inverter-based resources to rapidly mitigate local power imbalances upon events. The controller design is based directly on historical measurement sequences, and does not require identification of a parametric power system model. Theoretical results are provided to support the approach. Simulation studies on a nonlinear three-area test system demonstrate that the controller provides fast and localized frequency control under several types of contingencies.
Roughly speaking, systems and control theory deals with the problem of making a concrete physical system behave according to certain desired specifications. To achieve this desired behavior, the system can be interconnected with a physical device, called a controller. The problem of finding a mathematical description of such a controller is called the
control design problem
.
Physics aims to describe, classify, and predict natural phenomena, while engineering aims to design new or modify existing ones. A phenomenon is characterized by some observed variables.
Data‐based safe gain‐scheduling controllers are presented for discrete‐time linear parameter‐varying systems (LPV) with polytopic models. First, ‐contractivity conditions are provided under which the safety and stability of the LPV systems are unified through Minkowski functions of the safe sets. Then, a data‐based representation of the closed‐loop LPV system is provided, which requires less restrictive data richness conditions than identifying the system dynamics. This sample‐efficient closed‐loop data‐based representation is leveraged to design data‐driven gain‐scheduling controllers that guarantee ‐contractivity and, thus, invariance of the safe sets. It is also shown that the problem of designing a data‐driven gain‐scheduling controller for a polyhedral (ellipsoidal) safe set amounts to a linear program (a semi‐definite program). The motivation behind direct learning of a safe controller is that identifying an LPV system requires satisfying the persistence of the excitation (PE) condition. It is shown in this paper, however, that directly learning a safe controller and bypassing the system identification can be achieved without satisfying the PE condition. This data‐richness reduction is of vital importance, especially for LPV systems that are open‐loop unstable, and collecting rich samples to satisfy the PE condition can jeopardize their safety. A simulation example is provided to show the effectiveness of the presented approach.
In the context of data-driven control, persistence of excitation (PE) of an input sequence is defined in terms of a rank condition on the Hankel matrix of the input data. For nonlinear systems, recent results employed rank conditions involving collected input and state/output data, for which no guidelines are available on how to satisfy them a priori. In this letter, we first show that a set of discrete impulses is guaranteed to be persistently exciting for any controllable LTI system. Based on this result, for certain classes of nonlinear systems, we guarantee persistence of excitation of sequences of basis functions
a priori
, by design of the physical input only.
The fundamental lemma by Willems and coauthors enables a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays a key role in data-driven simulation and control. The fundamental lemma relies on a persistently exciting input to the system to ensure that the Hankel matrix of resulting input/output data has the “right” rank, meaning that its columns span the entire subspace of trajectories. However, such binary rank conditions are known to be fragile in the sense that a small additive noise could already cause the Hankel matrix to have full rank. In this paper we present a robust version of the fundamental lemma. The idea behind the approach is to guarantee certain lower bounds on the singular values of the data Hankel matrix, rather than qualitative rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of an input Hankel matrix is sufficiently large, inspiring a quantitative notion of persistency of excitation. We highlight the relevance of the result in a data-driven control case study by comparing the predictive control performance for varying degrees of persistently exciting data.
This survey is focused on certain sequential decision-making problems that involve optimizing over probability functions. We discuss the relevance of these problems for learning and control. The survey is organized around a framework that combines a problem formulation and a set of resolution methods. The formulation consists of an infinite-dimensional optimization problem. The methods come from approaches to search optimal solutions in the space of probability functions. Through the lenses of this overarching framework we revisit popular learning and control algorithms, showing that these naturally arise from suitable variations on the formulation mixed with different resolution methods. A running example, for which we make the code available, complements the survey. Finally, a number of challenges arising from the survey are also outlined.
The result of J.C. Willems et al. “A note on persistency of excitation”, System & Control Letters, 2005 gives identifiability conditions for system identification as well as data-driven representations for data-driven control. The existing proofs however are proofs by contradiction and do not give insight into the assumptions of controllability and persistency of excitation of the input. Moreover, the existing proofs do not clarify how conservative the assumptions are. We provide an alternative constructive proof for the single-input case. It is shown that persistency of excitation of order more than the time horizon is needed in nongeneric cases, corresponding to special initial conditions. The special initial conditions are explicitly characterized in terms of the solution of a Sylvester equation. Another contribution of the paper is a representation of a scalar persistently exciting input of a finite order as an output of an autonomous linear time-invariant system.
In this letter, we discuss the problem of optimal control for affine systems in the context of data-driven linear programming. First, we introduce a unified framework for the fixed point characterization of the value function,
-function and relaxed Bellman operators. Then, in a model-free setting, we show how to synthesize and estimate Bellman inequalities from a small but sufficiently rich dataset. To guarantee exploration richness, we complete the extension of Willems’ fundamental lemma to affine systems.
This letter develops a method to learn optimal controls from data for bilinear systems without a priori knowledge of the dynamics. Given an unknown bilinear system, we characterize when the available data is sufficiently informative to solve the optimal control problem. This characterization leads us to propose an online control experiment design procedure that guarantees that any input/state trajectory can be represented as a linear combination of collected input/state data matrices. Leveraging this representation, we transform the original optimal control problem into an equivalent data-based optimization problem with bilinear constraints. We solve the latter by iteratively employing a convex-concave procedure to find a locally optimal control sequence. Simulations show that the performance of the proposed data-based approach is comparable with model-based methods.
In a paper by Willems
et al.
, it was shown that persistently exciting data can be used to represent the input–output behavior of a linear system. Based on this fundamental result, we derive a parametrization of linear feedback systems that paves the way to solve important control problems using data-dependent linear matrix inequalities only. The result is remarkable in that no explicit system's matrices identification is required. The examples of control problems we solve include the state and output feedback stabilization, and the linear quadratic regulation problem. We also discuss robustness to noise-corrupted measurements and show how the approach can be used to stabilize unstable equilibria of nonlinear systems.
Classical linear time-invariant system simulation methods are based on a transfer function, impulse response, or input/state/output representation. We present a method for computing the response of a system to a given input and initial conditions directly from a trajectory of the system, without explicitly identifying the system from the data. Similarly to the classical approach for simulation, the classical approach for control is model-based: first a model representation is derived from given data of the plant and then a control law is synthesized using the model and the control specifications. We present an approach for computing a linear quadratic tracking control signal that circumvents the identification step. The results are derived assuming exact data and the simulated response or control input is constructed off-line.
We have examined an optimization approach to iterative control
design. The important ingredient is that the gradient of the design
criterion is computed from measured closed loop data. The approach is
thus not model-based. The scheme converges to a stationary point of the
design criterion under the assumption of boundedness of the signals in
the loop. From a practical viewpoint, the scheme offers several
advantages. It is straightforward to apply. It is possible to control
the rate of change of the controller in each iteration. The objective
can be manipulated between iterations in order to tighten or loosen
performance requirements. Certain frequency regions can be emphasized if
desired. This direct optimal tuning algorithm is particularly well
suited for the tuning of the basic control loops in the process
industry, which are typically PID loops. These primary loops are often
very badly tuned, making the application of more advanced (for example,
multivariable) techniques rather useless. A first requirement in the
successful application of advanced control techniques is that the
primary loops be tuned properly. This new technique appears to be a very
practical way of doing this, with an almost automatic procedure
Heymann's lemma is proved by a simple induction argument.
Recently, various algorithms for data-driven simulation and control have been proposed based on the Willems' fundamental lemma. However, when collected data are noisy, these methods lead to ill-conditioned data-driven model structures. In this work, we present a maximum likelihood framework to obtain an optimal data-driven model, the signal matrix model, in the presence of output noise. Data compression and noise level estimation schemes are also proposed to apply the algorithm efficiently to large datasets and unknown noise level scenarios. Two approaches in system identification and receding horizon control are developed based on the derived optimal estimator. The first one identifies a finite impulse response model. This approach improves the least-squares estimator with less restrictive assumptions. The second one applies the signal matrix model as the predictor in predictive control. The control performance is shown to be better than existing data-driven predictive control algorithms, especially under high noise levels. Both approaches demonstrate that the derived estimator provides a promising framework to apply data-driven algorithms to noisy data.
This paper formulates an input design approach for truncated infinite impulse response identification in the context of implicit model representations recently used as basis for data-driven simulation and control approaches. Precisely, the considered model consists of a linear combination of the columns of a data (or signal) matrix. An optimal combination for the case of noisy data was recently proposed using a maximum likelihood approach, and the objective here is to optimize the input entries of the data matrix such that the mean-square error matrix of the estimate is minimized. A least-norm problem is derived in terms of the optimality criteria typically considered in the experiment design literature. Numerical results showcase the improved estimation fit achieved with the optimized input.
We study the problem of finite-time constrained optimal control of unknown stochastic linear time-invariant systems, which is the key ingredient of a predictive control algorithm - albeit typically having access to a model. We propose a novel distributionally robust data-enabled predictive control (DeePC) algorithm which uses noise-corrupted input/output data to predict future trajectories and compute optimal control inputs while satisfying output chance constraints. The algorithm is based on (i) a non-parametric representation of the subspace spanning the system behaviour, where past trajectories are sorted in Page or Hankel matrices; and (ii) a distributionally robust optimization formulation which gives rise to strong probabilistic performance guarantees. We show that for certain objective functions, DeePC exhibits strong out-of-sample performance, and at the same time respects constraints with high probability. The algorithm provides an end-to-end approach to control design for unknown stochastic linear time-invariant systems. We illustrate the closed-loop performance of the DeePC in an aerial robotics case study.
In this paper we design suboptimal control laws for an unknown linear system on the basis of measured data. We focus on the suboptimal linear quadratic regulator problem and the suboptimal H2 control problem. For both problems, we establish conditions under which a given data set contains sufficient information for controller design. We follow up by providing a data-driven parameterization of all suboptimal controllers. We will illustrate our results by numerical simulations, which will reveal an interesting trade-off between the number of collected data samples and the achieved controller performance.
In this article, we propose a new method to obtain feedback controllers of an unknown dynamical system directly from noisy input/state data. The key ingredient of our design is a new matrix S-lemma that will be proven in this article. We provide both strict and nonstrict versions of this S-lemma, which are of interest in their own right. Thereafter, we will apply these results to data-driven control. In particular, we will derive nonconservative design methods for quadratic stabilization,
and
control, all in terms of data-based linear matrix inequalities. In contrast to previous work, the dimensions of our decision variables are independent of the time horizon of the experiment. Our approach, thus, enables control design from large datasets.
This paper considers the problem of designing state feedback data-driven controllers for nonlinear continuous-time systems. Specifically, we consider a scenario where the unknown dynamics can be parametrized in terms of known basis functions and the available measurements are corrupted by unknown-but-bounded noise. The goal is to use this noisy experimental data to directly design a rational state-feedback control law guaranteed to stabilize all plants compatible with the available information. The main result of the paper shows that, by using Rantzer’s Dual Lyapunov approach, combined with elements from convex analysis, the problem can be recast as an optimization over positive polynomials, which can be relaxed to a semi-definite program through the use of Sum-of-Squares and semi-algebraic optimization arguments. Three academic examples are considered to illustrate the effectiveness of the proposed method.
Willems
et al.
’s fundamental lemma asserts that all trajectories of a linear system can be obtained from a single given one, assuming that a persistency of excitation and a controllability condition hold. This result has profound implications for system identification and data-driven control, and has seen a revival over the last few years. The purpose of this letter is to extend Willems’ lemma to the situation where multiple (possibly short) system trajectories are given instead of a single long one. To this end, we introduce a notion of collective persistency of excitation. We will show that all trajectories of a linear system can be obtained from a given finite number of trajectories, as long as these are collectively persistently exciting. We will demonstrate that this result enables the identification of linear systems from data sets with missing samples. Additionally, we show that the result is of practical significance in data-driven control of unstable systems.
The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this paper, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data as data-driven control is possible only with data that are informative for system identification.
Control Theory for Linear Systems deals with the mathematical theory of feedback control of linear systems. It treats a wide range of control synthesis problems for linear state space systems with inputs and outputs. The book provides a treatment of these problems using state space methods, often with a geometric flavour. Its subject matter ranges from controllability and observability, stabilization, disturbance decoupling, and tracking and regulation, to linear quadratic regulation, H2 and H-infinity control, and robust stabilization. Each chapter of the book contains a series of exercises, intended to increase the reader's understanding of the material. Often, these exercises generalize and extend the material treated in the regular text.
Optimal input design for parameter estimation has obtained extensive coverage in the past. A key problem here is that the optimal input depends on some unknown system parameters that are to be identified. Adaptive design is one of the fundamental routes to handle this problem. Although there exist a rich collection of results on this problem, there are few results that address dynamical systems. This paper presents sufficient conditions for convergence/consistency and asymptotic optimality for a class of adaptive systems consisting of a recursive prediction error estimator and an input generator depending on the timevarying parameter estimates. The results apply to a general family of single input single output linear time-invariant systems. An important application is adaptive input design for which the results imply that, asymptotically in the sample size, the adaptive scheme recovers the same accuracy as the off-line prediction error method that uses data from an experiment where perfect knowledge of the system has been used to design an optimal input spectrum.
In this work, we study the design of a controller using system data. We present three data-driven approaches based on the notion of control as interconnection. In the first approach, we use both the data and representations to compute control variable trajectories that impose a prescribed path on the to-be-controlled variables. The second method is completely data-driven and we prove sufficient conditions for determining a controller directly from data. Finally, we show how to determine a controller directly from data in the case where the control and to-be-controlled variables coincide.
Identification of systems operating in closed loop is an important problem in industrial applications, where model-based control is used to an increasing extent. For model-based controllers, plant changes over time eventually result in a mismatch between the dynamics of any initial model in the controller and the actual plant dynamics. When the mismatch becomes too large, control performance suffers and it becomes necessary to re-identify the plant to restore performance. Often the available data are not informative enough when the identification is performed in closed loop and extra excitation needs to be injected. This paper considers the problem of generating such excitation with the least possible disruption to the normal operations of the plant. The methods explicitly take time domain constraints into account. The formulation leads to optimal control problems which are in general very difficult optimization problems. Computationally tractable solutions based on Markov decision processes and model predictive control are presented. The performance of the suggested algorithms is illustrated in two simulation examples comparing the novel methods and algorithms available in the literature.
Filtering and system identification are powerful techniques for building models of complex systems. This 2007 book discusses the design of reliable numerical methods to retrieve missing information in models derived using these techniques. Emphasis is on the least squares approach as applied to the linear state-space model, and problems of increasing complexity are analyzed and solved within this framework, starting with the Kalman filter and concluding with the estimation of a full model, noise statistics and state estimator directly from the data. Key background topics, including linear matrix algebra and linear system theory, are covered, followed by different estimation and identification methods in the state-space model. With end-of-chapter exercises, MATLAB simulations and numerous illustrations, this book will appeal to graduate students and researchers in electrical, mechanical and aerospace engineering. It is also useful for practitioners. Additional resources for this title, including solutions for instructors, are available online at www.cambridge.org/9780521875127.
Model Predictive Control (MPC) is a well known and widely used advanced optimal control technique. It is a common practice to use a process model to predict the future behavior of the plant. The model/plant mismatch may have direct consequences on the quality of the prediction, causing potential controller performance degradation. An approach to tackle this problem may be to use closed loop system identification such that the model parameters are estimated and updated online, while the feedback controller is running. This adaptation requires that the plant input is persistently exciting, however a standard MPC controller is not able to provide sufficient input frequency content to obtain reliable parameter estimates. In this article a Persistently Exciting Model Predictive Control (PE-MPC) formulation is given. A Finite Im-pulse Response (FIR) model is adopted for prediction, and Recursive Least Square (RLS) is used for parameter estimation. Moreover, it is shown how to derive a persistently exciting constraint, suitable for implementation with MPC. Finally, it is explained how to implement the optimiza-tion problem such that, every sample time, only two Quadratic Programming (QP) problems are solved, and the optimal solution is applied in a receding horizon fashion. In the final part of the work, a simulation based example is given to show the effectiveness of the approach.
The purpose of the design of identification experiments is to make the collected data maximally informative with respect to the intended use of the model, subject to constraints that might be at hand. When the true system is replaced by an estimated model, there results a performance degradation that is due to the error in the transfer function estimates. Using some recent asymptotic expressions for the bias and the variance of the estimated transfer function, it is shown how this performance degradation can be minimized by a proper experiment design. Several applications, where it is beneficial to let the experiment be carried out in closed loop, are highlighted.
New algorithms for identification of a balanced state space representation are proposed. They are based on a procedure for the estimation of impulse response and sequential zero input responses directly from data. The proposed algorithms are more efficient than the existing alternatives that compute the whole Hankel matrix of Markov parameters. It is shown that the computations can be performed on Hankel matrices of the input–output data of various dimensions. By choosing wider matrices, we need persistency of excitation of smaller order. Moreover, this leads to computational savings and improved statistical accuracy when the data is noisy. Using a finite amount of input–output data, the existing algorithms compute finite time balanced representation and the identified models have a lower bound on the distance to an exact balanced representation. The proposed algorithm can approximate arbitrarily closely an exact balanced representation. Moreover, the finite time balancing parameter can be selected automatically by monitoring the decay of the impulse response. We show what is the optimal in terms of minimal identifiability condition partition of the data into “past” and “future”.
All approaches to optimal experiment design for control have so far focused on deriving an input signal (or input signal spectrum) that minimizes some control-oriented measure of plant/model mismatch between the nominal closed-loop system and the actual closed-loop system, typically under a constraint on the total input power. In practical terms, this amounts to finding the (constrained) input signal that minimizes a measure of a control-oriented model uncertainty set. Here we address the experiment design problem from a “dual” point of view and in a closed-loop setting: given a maximum allowable control-oriented model uncertainty measure compatible with our robust control specifications, what is the cheapest identification experiment that will give us an uncertainty set that is within the required bounds? The identification cost can be measured by either the experiment time, the performance degradation during experimentation due to the added excitation signal, or a combination of both. Our results are presented for the situation where the control objective is disturbance rejection only.
We prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This is then applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The related question of when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts is of central importance in subspace system identification.
An optimal experiment design for system identification is studied.
The main contribution is the development of an adaptive method for the
direct design of FIR filters for the input spectrum design problem. The
accuracy of the identified model is measured in terms of the closed-loop
performance of the system using the controller designed from the model.
Under the assumption that the identified parameters are sufficiently
close to their true values, we show that this problem may be formulated
as a convex optimization problem with linear matrix inequality
constraints. Thus, a global solution (if feasible) is guaranteed and the
solution may further achieve any demanded accuracy. The problem
formulation is particularly suited for a practical implementation, thus
the extension of the experiment design problem into an
iterative/adaptive identification-experiment design framework is
straight forward. The adaptive approach is further studied in a
simulation example, where the rapid convergence of the method is noted,
and the superior result compared to an arbitrary experiment design is
clear. The example support the use of the approximations taken in the
theoretical approach
A framework for reformulating input design problems in prediction error identification as convex optimization problems is presented. For linear time-invariant single input/single output systems, this framework unifies and extends existing results on open-loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates. Basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided. A wide range of model quality constraints can be handled. In particular, different frequency-by-frequency constraints can be used. This opens up new applications of input design in areas such as robust control. Furthermore, quality specifications can be imposed on all models in a confidence region. Thus, allowing for statements such as "with at least 99% probability the model quality specifications will be satisfied".
Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs, as well as dual optimization problems, can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems.
The family of m -input, n -dimensional linear systems can be globally identified with a generic input sequence of length 2mn . This bound is the best possible. A best bound is provided also for a corresponding local identification problem.
Data-driven system level synthesis
- A Xue
- N Matni
From noisy data to feedback controllers: Non-conservative design via a matrix S-lemma
- van waarde