Output feedback control for unknown LTI systems driven by unknown periodic disturbances

doi:10.1016/j.automatica.2018.10.020

Automatica

Volume 99, January 2019, Pages 112-119

https://doi.org/10.1016/j.automatica.2018.10.020 Get rights and content

Abstract

This paper considers unknown minimum-phase LTI systems with known relative degree and system order. The main aim is to reject the unknown, unmatched sinusoidal disturbances and make the output track a given trajectory with the output feedback. The essence of the control design is composed of disturbance parametrization, K-filter technique and adaptive backstepping procedure. Firstly, the unmeasured system states are represented in terms of filtered input and output signals. Then, the disturbance information in the output signal is parametrized and the problem is converted to an adaptive control problem. After that, the K-filter approach is employed to redefine the system states that enable to use a backstepping technique. An adaptive output feedback controller is designed recursively. It is proven that the equilibrium at the origin is globally uniformly stable and the output signal tracks the reference signal asymptotically. Finally, the effectiveness of the controller is illustrated with a numerical simulation. The robustness of the closed loop system with respect to an additive unmodelled noise is also discussed.

Introduction

The problem of sinusoidal disturbance cancellation is observed in many real-world applications such as active suspension control (Landau, Constantinescu, & Rey, 2005), active noise control (Bodson, Jensen, & Douglas, 2001) and marine vehicles Basturk and Krstic, 2013b, Marconi et al., 2002. One of the techniques to approach this problem in linear systems is the internal model principle Francis and Wonham, 1975, Johnson, 1971. This principle suggests the disturbance to be written as the output of a known linear dynamic system (the so-called exosystem). In order to compensate for the disturbance, a reduplicated model of this exosystem is added to the feedback loop. Since the disturbance rejection with full-state feedback may not be feasible in practical applications, the studies have focused on developing control methods with output feedback.

The problem of disturbance rejection for linear systems by output feedback may be divided into four categories. The first one is a basic case in which the system parameters and the exosystem are known. This problem is studied in Davison (1976) and Francis, Bruce, and Wonham (1976). In the second category, the system parameters are known but the disturbance is the output of an uncertain exosystem. This problem is solved for a stable plant of arbitrary relative degree with one matched sinusoidal disturbance in Bobtsov and Pyrkin (2009). Multisinusoidal disturbance compensation is achieved for a system with arbitrary relative degree in Marino and Tomei (2013). The same problem is studied in Kim and Shim (2015) with an assumption that the zero dynamics of the plant is hyperbolic, which means that there is no zero on the imaginary axis of the complex plane. The systems in Marino and Tomei (2013) and Kim and Shim (2015) are not restricted to be stable or minimum phase. Moreover, the problem of rejecting harmonic disturbances acting on the output signal is addressed in Serrani (2006). In the third case, which assumes that the system is uncertain whereas the exosystem is known, an adaptive algorithm is introduced for stable systems to reject the unmatched disturbance with known frequencies in Marino and Tomei (2015) and Pigg and Bodson (2010).

The fourth, and the most complex case is the one in which the system parameters are unknown and the sinusoidal disturbance is generated by an uncertain exosystem. This is the one that we consider in this paper. For a minimum phase system whose relative degree is higher than unity, an iterative algorithm is suggested in Bobtsov, Kolyubin, Kremlev, and Pyrkin (2012) to compensate for a single-frequency sinusoidal disturbance. The case of multiharmonic disturbance and arbitrary relative degree is studied in Bobtsov, Kolyubin, and Pyrkin (2013). Moreover, assuming that the system is minimum phase with known relative degree, a design of an adaptive learning regulator is introduced in Marino and Tomei (2011). In Marino and Tomei (2016), a biased multi-sinusoidal disturbance compensator is developed for a stable system of an unknown order and an unknown relative degree. The rejection of multisinusoidal disturbances is studied in Isidori and Marconi (2013) for the systems which have multiple zeros at the origin. Adaptive control of unknown linear systems is dealt with state-derivative feedback in Basturk and Krstic (2013a) and Basturk and Krstic (2014).

This study contributes mainly to the fourth case. In Bobtsov et al. (2012), the cancellation of a single-frequency disturbance is considered. The researchers in Bobtsov et al. (2013) drive the output to zero under matched multiharmonic disturbances, assuming a lower bound for frequencies. The adaptive learning regulator in Marino and Tomei (2011) assumes that the unknown parameters are in a known bounded region. The disturbance compensator in Marino and Tomei (2016) is restricted to stable linear systems. The assumption made in Isidori and Marconi (2013) is that the relative degree is unitary and the initial states of the exosystem are in a compact set. In this paper, we consider all of these challenges simultaneously.

We consider uncertain and minimum-phase LTI systems driven by unknown and unmatched sinusoidal disturbances. We assume that the followings are known: (i) upper bound of the plant order, (ii) upper bound of the exosystem order, (iii) relative degree of the plant, and (iv) the sign of the high frequency gain. We propose an algorithm to reject the disturbances and to make the output track a reference trajectory with the output feedback. We show that the equilibrium at the origin is globally uniformly stable. We also make a minor robustness analysis of the closed loop system with respect to an additive unmodelled noise.

The problem definition is stated in Section 2. The problem is reformulated in Section 3. Adaptive controller design is introduced in Section 4. In Section 5, the stability theorem and its proof are presented. Section 6 shows the feasibility of the proposed controller using simulation examples.

Notations: We use $e_{i}$ as a column vector whose $i$ th element is $1$ and the rest are $0$ . The parameter estimation and estimation errors are denoted with the symbols “ $\hat{}$ ” and “ $\tilde{}$ ”, respectively. As an example, estimation error of $θ$ state is $\tilde{θ} = θ - \hat{θ}$ where $\hat{θ}$ is the estimation of $θ$ . The vector $0_{n}$ shows ${[0, \dots, 0]}^{T} \in R^{n} .$ The magnitude and phase angle are denoted by $| \cdot |$ and $∠$ , respectively. A random signal $χ$ , which is normally distributed with mean $\bar{μ}$ and variance ${\bar{σ}}^{2}$ , is denoted by $χ \sim N (\bar{μ}, {\bar{σ}}^{2})$ . Argument $^{'} t^{'}$ is omitted when it comes alone.

Section snippets

Problem statement

We consider the following single input single output system: $\dot{x} = A x - a y + [\begin{matrix} 0_{n - m - 1} \\ b \end{matrix}] u + d ν$ $y = e_{1}^{T} x + d_{n} ν,$ where $A = [\begin{matrix} 0 \\ ⋮ & I_{n - 1} \\ 0 & \dots & 0 \end{matrix}], a = [\begin{matrix} a_{n - 1} \\ ⋮ \\ a_{1} \\ a_{0} \end{matrix}], b = [\begin{matrix} b_{m} \\ ⋮ \\ b_{1} \\ b_{0} \end{matrix}], d = [\begin{matrix} d_{n - 1} \\ ⋮ \\ d_{1} \\ d_{0} \end{matrix}],$ the state $x = {[x_{1}, \dots, x_{n}]}^{T} \in R^{n}$ , the input $u \in R$ , the output $y \in R$ . The disturbance is given by $ν = g_{0} + \sum_{i = 1}^{q} g_{i} cos (ω_{i} t + ϕ_{i}),$ where $g_{0}, g_{i}, ω_{i}, ϕ_{i} \in R .$ The plant parameters $a, b, d, d_{n}$ and disturbance parameters $g_{0}, g_{i}, ω_{i}, ϕ_{i}$ for $i = 1, \dots, q$ are unknown. The disturbance $ν$ is not measured. The only measured signal is $y$ .

Remark 1

We consider a general case where all of the system states are

Reformulation of the problem

In this section, we represent the system in terms of filtered input and output signals. This representation is used for unmeasured states in the output dynamics. We extract the information of the unknown disturbance in the output dynamics and parametrize it using the technique given in Nikiforov (2004). Finally, we employ the Kreisselmeier filters (K-filters), which were initially suggested in Kreisselmeier (1977), to prepare a ground for a backstepping procedure. The disturbance

Adaptive controller design

In this section, we apply a backstepping technique to design an adaptive controller for (48)–(51). We first define new error terms to get virtual control input for backstepping, i.e. stabilizing functions. Then, performing a recursive design, we propose an actual control input with the update laws, which achieves asymptotic tracking of $y_{r}$ by $y$ . The filter design and controller design procedure are schematically given in Fig. 1.

We employ the following coordinate transformation: $z_{1} = y - y_{r}$ $z_{i} = v_{m, i} - \hat{ϱ} y$

Stability analysis

The main theorem is stated as follows.

Theorem 1

Consider the closed-loop system consisting of the plant (1) – (2) , unknown disturbance (4) , the disturbance observer filters (31) – (34) , K-filters (43) – (47) , the control law (57) and the parameter update laws (58) – (61) . Under Assumption 1, Assumption 2, Assumption 3, Assumption 4, Assumption 5 , the signals $\hat{ϱ}, y \in R, η, v_{m}, λ, ϵ_{i n}, \bar{x}, x \in R^{n}, z \in R^{ρ}, ε, Z, Ψ, Φ \in R^{2 q + 1}, φ \in R^{1 \times (2 q + 1) (n + m + 1)}, \hat{θ} \in R^{(2 q + 1 + (n + m + 1) (2 q + 2))}$ are globally bounded and the asymptotic tracking is

Numerical simulations

In this section, we present two simulations. Case I shows the effectiveness of the controller. Case II illustrates robustness of the closed loop system with respect to an additional unmodelled noise. We consider the following unstable relative-degree-two plant: $\dot{x} = [\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}] x - [\begin{bmatrix} 0.6 \\ 0.5 \\ 0.4 \end{bmatrix}] y + [\begin{bmatrix} 0 \\ 1 \\ 0.8 \end{bmatrix}] u + [\begin{bmatrix} 0 \\ 0 \\ 0.9 \end{bmatrix}] ν$ $y = e_{1}^{T} x .$ In Case I, we choose the reference signal $y_{r}$ and the unknown disturbance $ν$ as given in Table 2. The number of distinct frequencies, $q$ , is assumed to be 2 in the design. We choose initial

Conclusion

In this study, we solve the problem of an unknown sinusoidal disturbance rejection for unknown and minimum phase LTI systems with the output feedback. Firstly, the system states are represented as a linear function of system parameters by filtering input and output signals. Then, we parametrize the disturbance information in the output signal. By employing K-filter technique, we reform the system as a new system whose states are available so that we approach the problem as a backstepping based

Cemal Tugrul Yilmaz received his B.S. and M.S. degrees in Mechanical Engineering from Bogazici University, Turkey in 2015 and 2018. His main research interests are adaptive control, disturbance rejection, state estimation and control of time-delay systems.

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Halil Ibrahim Basturk received his B.S. and M.S. degrees in Mechanical Engineering from Bogazici University in 2006 and 2008, and the Ph.D. degree in Mechanical and Aerospace Engineering from the University of California at San Diego in 2013. Since 2014 he has been Assistant Professor of Mechanical Engineering at Bogazici University, Istanbul, Turkey. His research interests include disturbance estimation/cancellation, adaptive control, boundary control, and control of delay systems.

^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Martin Guay under the direction of Editor Miroslav Krstic.

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Brief paperOutput feedback control for unknown LTI systems driven by unknown periodic disturbances☆

Abstract

Introduction

Section snippets

Problem statement

Reformulation of the problem

Adaptive controller design

Stability analysis

Numerical simulations

Conclusion

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IFAC Proceedings Volumes

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An iterative algorithm of adaptive output control with complete compensation for unknown sinusoidal disturbance

Automation and Remote Control

Compensation of unknown sinusoidal disturbances in linear plants of arbitrary relative degree

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Active noise control for periodic disturbances

IEEE Transactions on Control Systems Technology

Brief paper
Output feedback control for unknown LTI systems driven by unknown periodic disturbances☆