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Observer based minimum variance control of uncertain piecewise affine systems subject to additive noise
Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Observer based minimum variance control of uncertain piecewise affine systems subject to additive noise Hamidreza Razavi, Kaveh Merat, Hassan Salarieh ∗ , Aria Alasty, Ali Meghdari Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Azadi Ave, Tehran, Iran
article
info
Keywords: Hybrid observer–controller Non-vanishing additive noise Stochastic hybrid systems Piecewise quadratic Lyapunov function
abstract In this paper, a class of uncertain piecewise affine (PWA) systems, subject to system and measurement additive noises is studied. The additive noise signals considered here do not vanish at the equilibrium and the uncertainties are norm bounded. The problem of minimizing the bound on the variance of the steady response of uncertain PWA systems, by means of a hybrid observer–controller, is formulated as an optimization problem subject to a number of constraints in the form of matrix inequalities. The derived constraints are obtained by considering a piecewise quadratic Lyapunov function in combination with the general stability conditions regarding the existence of an upper stochastic bound on the steady state variance for a class of stochastic hybrid systems (SHS). Then the uncertain PWA approximation of a practical system with nonlinear dynamics is presented considering system and measurement noises. The uncertainties arise in the form of the difference between the actual nonlinear dynamics and the PWA approximation. Utilizing the introduced methods, a hybrid observer–controller is designed and implemented on the nonlinear system to demonstrate the effectiveness of the proposed controller design procedure. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Piecewise affine (PWA) systems are a powerful modeling tool for the control of nonlinear systems. These systems constitute a class of hybrid systems and are often encountered in control systems where piecewise-linear components such as dead-zone, saturation, relays and hysteresis are present. PWA systems may also be used to approximate the dynamics of other nonlinear systems. The stability analysis of deterministic PWA systems based on the multiple Lyapunov function method was studied by many authors [1–3]. Also, the state feedback control of PWA systems has attracted many researches [4,5,3] which consider deterministic systems controlled by continuous-time controllers. The simultaneous observer and controller design for deterministic PWA systems was studied by Rodrigues and How [6]. The PWA systems considered here however, are more generalized; as they include both norm bounded uncertainty terms and additive noise terms in both the process and measurement equations. Many practical systems, including PWA systems, are subject to uncertainties due to modeling error and external disturbances. In the case of using a PWA system to model the behavior of a nonlinear system, the PWA system will be subjected to more approximation error, thus increasing the uncertainty in the model. Several approaches are proposed for the controller design of uncertain PWA systems. Ben Abdallah et al. [7] assumed that the uncertain system parameters belong to a convex polyhedral and utilized the results obtained by Oliveira and Peres [8] and Xu and Xie [9]. It was proved that obtaining a solution that satisfies the related matrix inequality constraints at the vertices of the mentioned convex polyhedral will result
∗
Corresponding author. Tel.: +98 21 6616 5538; fax: +98 21 6600 0021. E-mail address: [email protected] (H. Salarieh).
http://dx.doi.org/10.1016/j.nahs.2015.09.002 1751-570X/© 2015 Elsevier Ltd. All rights reserved.
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in obtaining the desired controller gain values and piecewise quadratic Lyapunov function (PQLF) ensuring the closed loop system stability. However, as the number of system states, partitions and uncertainty polyhedral vertices increase, obtaining a solution by this method becomes exponentially difficult. Feng [10] proposed a method in which only the upper bounds of the system uncertainties are included in the inequalities. Zhang et al. [11] modeled the system uncertainties similar to Feng [10]. The authors further generalized the model by including a one dimensional stochastic noise term that vanishes at the steady state equilibrium point. The piecewise affine controller design problem was then formulated as an optimization problem subject to a number of matrix inequality constraints in order to achieve stochastic stability in the mean square. However this design has limited applicability as in practice the noise terms do not usually vanish at the steady state equilibrium point and, as will be discussed later, achieving stochastic stability in the mean square is impossible. The inclusion of the additive noise terms in the PWA systems studied here makes them a subset of the more generalized stochastic hybrid systems (SHS). The SHSs comprise a vast range of systems where some randomness is introduced to a hybrid model, with both discrete and continuous states. The randomness in the SHS could be considered in both the evolution of the discrete state and dynamics of the continuous states. Bujorianu and Lygeros [12] proposed a general structure for SHS where the continuous evolution is defined by a stochastic diffusion process and the governing relation for the discrete state is probabilistic. In this framework, both discrete and continuous dynamics could be interconnected, in other words both of them could be dependent on the discrete and continuous states. Given that the design of controller and observer for the general SHS is very challenging in its general format, many methods have been studied that consider a special class of general SHS for stability analysis and controller design. The Lyapunov based stability theorems for different classes of stochastic hybrid systems have been widely investigated. Dimarogonas and Kyriakopoulos [13] gave stochastic stability theorems for a class of SHS with arbitrary change of discrete state. The proposed theorems are based on conditions that guarantee the stability of the steady state equilibrium point with probability one regardless of the discrete state dynamics and with noise that vanishes at the steady state equilibrium point. It is notable that discarding the discrete dynamics can simplify the conditions but may result in very conservative sufficient conditions. The asymptotic stability condition for SHS with a Markov process as the discrete evolution dynamics is investigated by Abate et al. [14] where it is assumed that the jumps have probabilistic dwell-time in each discrete state. More recently Xiong et al. [15] derived the stability condition, in the form of Linear Matrix Inequalities (LMI), for switching linear systems. For a class of stochastic hybrid nonlinear systems with Markov jumps, the almost sure exponential stability theorem is given by Yuan and Lygeros [16]. Again there is a condition that requires the noise term to disappear at the steady state equilibrium point. It is remarkable that exponential and asymptotic stability, or stability with probability one, are not achievable in the presence of noise that remains non-zero at the steady state equilibrium point. Moving on to the controller design for SHS, Dong and Sun [17] have introduced the impulsive output feedback control for a class of nonlinear SHS with Markov jumps. Boukas [18] addressed the static output feedback control for an uncertain singular linear stochastic system with Markov switches, where there are norm bounded uncertainties and the noise signals vanish at the equilibrium. Another output feedback control design via LMI for continuous-time Markovian jump linear systems is proposed by De Farias et al. [19], which again guarantees the exponential stochastic mean square stability. Most of the literature for SHS stability and control design have not considered state dependent jumps and additive noises that do not vanish in the equilibrium point. Both of the aforementioned properties are required for the stochastic uncertain PWA systems considered here. In the case addressed in this article, asymptotic stability for mean square variance is impossible and only a bound on the second stochastic moment could be obtained. Similarly Rodrigues and Gollu [20] studied the control of a piecewise stochastic system with additive noise, but the common Lyapunov function method is applied and it is assumed that the drift and noise terms are continuous at the boundaries between subsystems and also no upper bound on the variance is given. One of the other close studies in the stability of stochastic hybrid systems with state-dependent switching is given by Wu et al. [21]. The stability theorem approach given is based on the generalization of Barbalat lemma to the stochastic hybrid systems. But still the given stability theorems consider vanishing noise at equilibrium which results in mean square stability. The state estimation for SHS has also been the topic of many researches. Battilotti [22] proposed a novel observer for SHS with Markov jumps satisfying some dwell-time. This observer guarantees that the state estimation error of the switching dynamics converges asymptotically to zero with probability one. Ahrens and Khalil [23] developed high-gain observers for a class of nonlinear systems with measurement noise. So there is no discrete state in the main system but the observer jumps between two gain values. Le Ny et al. [24] proposed scheduling Kalman filters for the observation of a class of stochastic linear systems where the Kalman gains are obtained via LMI. Most of the studies in observer design, consider Markov jumps in order to simplify the stability analysis. In this article however, state dependent jumps are considered. To the best of the authors knowledge however, the problem of stabilizing an uncertain stochastic PWA system by means of a piecewise affine observer–controller has never been addressed in the literature. Thus in this article, a systematic method is proposed for simultaneous design of observer and controller for uncertain stochastic PWA systems. In this system, the noises are additive and non-vanishing at the steady state equilibrium point, so the design procedure gives an upper bound on the combined estimation error and controller error variances. On the other hand, as stated before, the uncertainties in the considered PWA system may arise when the PWA system is utilized to model the behavior of a nonlinear system. If a control law for the nonlinear system is to be designed based
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on its approximated PWA system, the upper bound of the uncertainties in the PWA system should be known. Accuracy in the calculation of these upper bounds is crucial as they directly affect the stability of the closed loop system. However, the method of obtaining these upper bounds is seldom addressed in the literature. Thus another contribution of this article is to propose a systematic method to calculate the upper bounds of the uncertainties in the PWA system. The rest of the paper is organized as follows. In the next section, some preliminary definitions and properties are given about a general class of SHS alongside the conditions of their stability. Then in the following section, the design procedure for the combined controller and observer for uncertain stochastic PWA systems is introduced. Next, the proposed design method is implemented on a simplified nonlinear helicopter model to show its performance. The conclusions are drawn in the final section. 2. Preliminaries and hybrid stochastic stability The continuous state space of hybrid systems studied here is partitioned into a number of closed (possibly unbounded) polyhedral cells Ωi , i ∈ I = {1, 2, . . . , N }, such that ∪i∈I Ωi = ℜn . The index set of the cells containing the steady state equilibrium point (Here considered to be located at the origin) is denoted by I0 ⊆ I while the index set of the cells not containing the origin is denoted by I1 ⊆ I. Denoting the state vector of the system by x ∈ ℜn , the augmented state vector is defined by x¯ = xT defined by:
T
1 . Now given the fact that each cell is a polyhedral, the ith cell Ωi , enclosed by Ni facets, may be
Ωi = x|Ei x + ei = E¯ i x¯ ≥ 0 (1) where E¯ i = Ei ei , Ei ∈ ℜNi ×n is a constant matrix, each of whose rows is vector perpendicular to one of the Ni facets enclosing Ωi and ei ∈ ℜNi . Any two cells sharing a common facet are called level-1 neighboring cells. It is assumed that over the common boundary between two adjacent cells, the state of the system follows the dynamic equations of the cell with the lesser index number. A parametric description of the common facet between two level-1 neighboring cells, Ωi and Ωj , may be obtained as [6]:
Ωi ∩ Ωj ⊆ x|x = Fij s + lij , s ∈ ℜn−1 = x|τijT x − lij = 0
(2)
where Fij ∈ ℜn×(n−1) is a constant full rank matrix whose columns comprise a base mapping for the common facet between Ωi and Ωj . lij ∈ ℜn may be chosen as any arbitrary constant point on the common facet and τij ∈ ℜn represents the vector perpendicular to the common facet. A family of stochastic hybrid systems in Ito form, with state dependent jumps, is defined as: dx = fi (x) dt + gi (x) dw,
x ∈ Ωi ,
i∈I
(3)
where x ∈ ℜ is the state variable vector and w (t ) is a q-dimensional normalized Wiener process satisfying E [dw ] = 0 and E (dw) (dw)T = Iq dt with Iq as the q- dimensional identity matrix. fi (x) : ℜn → ℜn and gi (x) : ℜn → ℜn×q for ∀i ∈ I are bounded and Lipschitz continuous to ensure the existence and uniqueness of the corresponding solution process in Ωi . n
Definition 1 ([25]). The system given in (3) is stochastically second moment stable with bound d > 0 and symmetric weight matrix Q > 0 if starting from any initial condition we have: lim E x (t )T Qx (t ) < d.
(4)
t →∞
It is notable that this definition is different from stochastic stability in the mean-square defined in [26,11] as it does not require the variances to tend to zero in time which is only possible when there are multiplicative noises in the system. Definition 2. For each stochastic subsystem in (3) the local infinitesimal generator acting on a twice continuously differentiable scalar function Y ∈ C 2 : ℜn → ℜ is defined by: Li Y (x) =
1 ∂ 2 Y ( x) ∂ Y ( x) fi (x) + trace giT (x) g x . ( ) i ∂x 2 ∂ x2
(5)
Lemma 1 ([25]). Consider the hybrid stochastic system in (3). Let there exist twice continuously differentiable functions Vi : Ωi → ℜ+ (Multi-Lyapunov functions), constant d > 0, and scalar semi-positive (non-negative) function h (x) such that: Vi (x) = Vj (x) ,
x ∈ Ωi ∩ Ωj
Li Vi (x) ≤ −h (x) + d,
x ∈ Ωi ,
(6a) i ∈ I.
(6b)
Then we have: lim E [h (x)] ≤ d.
t →∞
(7)
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Remark 1 ([25]). A system is called Feller process if for all of its solution x (t ) and all bounded and continuous functions f (x), we have continuity for E [f (x)]. It should be noted that with Lipschitz continuity of fi and gi , one can prove the Feller property for the system in (3) using an approach like that of [27]. As mentioned in [28], if the conditions of Lemma 1 are met and the system is a Feller process, then there exists an invariant probability measure for the system. So we may conclude that the system is ergodic and it has steady distribution. Remark 2. In the current study the symbol > (≥) is used to denote matrix inequalities while the symbol ≻ is used to denote element-wise inequalities. Lemma 2 ([29]). For arbitrary matrices X , Y and scalar number ε > 0 we have:
ε X T X + ε −1 Y T Y ≥ X T Y + Y T X .
(8)
3. Observer–controller design for stochastic uncertain piecewise affine systems As a special case for (3), the following Ito diffusion formula may now be introduced to illustrate the dynamic equations of an uncertain stochastic piecewise affine system in each cell: dx = ((Ai + 1Ai ) x + (Bi + 1Bi ) u + ai + 1ai ) dt + Si dw, dz = ((Ci + 1Ci ) x + ci + 1ci ) dt + Ri dv,
x ∈ Ωi
x ∈ Ωi
i∈I
i∈I
(9a) (9b)
where Ai ∈ ℜn×n , ai ∈ ℜn , Bi ∈ ℜn×m , Ci ∈ ℜl×m and ci ∈ ℜl are the state matrix, system affine term, input matrix, output matrix and output affine term of the ith nominal local stochastic linear model, respectively. u ∈ ℜm is the system input vectorand Si dw defines the system additive white noise where w (t ) is a q-dimensional normalized Wiener process satisfying E (dw) (dw)T = Iq dt with Iq as the q-dimensional identity matrix. dz ∈ ℜl is the system output vector and is perturbed by the measurement additive white noise, Ri dv . v (t ) is a p-dimensional normalized Wiener process satisfying E [dv ] = 0 and E (dv) (dv)T = Ip dt with Ip as the p-dimensional identity matrix and Ri ∈ ℜn×p . 1Ai , 1ai , 1Bi , 1Ci and 1ci denote the uncertainty terms in Ai , ai , Bi , Ci and ci , respectively; and are of similar dimensions to that of their corresponding nominal values. It is assumed that w (t ) and v (t ) are uncorrelated. Utilizing the augmented state vector x¯ = xT
T
1 , (12) may be rewritten in the following form: A¯ i + 1A¯ i x¯ + B¯ i + 1B¯ i u dt + S¯i dw,
x ∈ Ωi i ∈ I ¯ ¯ dz = Ci + 1Ci x¯ dt + Ri dv, x ∈ Ωi i ∈ I T T where B¯ i = BTi 0 , C¯ i = Ci ci , S¯i = SiT 0 and A¯ i is defined by: A i ai A¯ i = dx¯ =
0
(10a) (10b)
0
and the uncertainty matrices 1A¯ i , 1B¯ i and 1C¯ i are augmented similar to A¯ i , B¯ i and C¯ i , respectively. Assumption 1. In order to present stability results, it is assumed that the upper bounds of the system uncertainties are known a priori, that is,
T 1ai 1Ai T 1ci 1Ci
1Ai 1Ci
1BTi 1Bi
≤
2 1ai ≤ E¯ iA 1ci ≤ E¯ iC2
(11a) (11b)
T EiB EiB
(11c)
2 where EiB is of the same dimensions as 1Bi and E¯ iA ∈ ℜ(n+1)×(n+1) and E¯ iC2 ∈ ℜ(n+1)×(n+1) are symmetric matrices. It is notable that the definition of the uncertainty upper bound matrices, for the system and output matrices, introduced in (11a) and (11b) do not require the uncertainty upper matrices to be decomposable in the format introduced by Feng in [10] and is thus slightly more general.
The purpose of this study is the second moment stabilizing of the system in (9) by means of a piecewise affine observer–controller of the following form: u = Ki xˆ + ki ,
x ∈ Ωi
i∈I
(12)
where Ki ∈ ℜ , ki ∈ ℜ are the constant gain matrix and offset term of the controller assigned to the ith nominal local stochastic linear model, respectively. xˆ ∈ ℜn is the estimation of the state vector x and satisfies the following Ito diffusion formula: m×n
m
dxˆ = Ai xˆ + Bi u + ai dt + Li dz − Ci xˆ + ci dt ,
x ∈ Ωi ,
i∈I
(13)
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where Li ∈ ℜn×l is the observer gain matrix of the ith nominal local stochastic linear model. Defining e = x − xˆ as the observer error and utilizing the augmented state vector x¯ = following form: u = K¯ i x¯ − Ki e, where K¯ i = Ki
x ∈ Ωi ,
xT
T
1 , the control law in (12) may be rewritten in the (14)
i∈I
ki . Now combining (10), (13) and (14) results in:
A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i 1A˜ i − Li 1C¯ i + 1Bi K¯ i
− B¯ i + 1B¯ i Ki ¯ S¯ 0 x ∈ Ωi , i ∈ I x¯ dt + i dw, ¯ (15) S i −L i R i Ai − Li Ci − 1Bi Ki T T ¯ = w T v T is an augmented (q + p)-dimensional normalized Wiener where 1A˜ = 1Ai 1ai , x¯¯ = x¯ T eT and w T T process. Given the augmented state vector x¯¯ = x¯ eT , a parametric description of the common facet between two level-1 neighboring cells, Ωi and Ωj may be obtained as: (16) Ωi ∩ Ωj ⊆ x¯¯ |x¯¯ = F¯ij s¯ + ¯lij , s¯ ∈ ℜ2n−1 = x¯¯ |τ¯ijT x¯¯ − ¯lij = 0 T T T T where ¯lij = lij 1 01×n , τ¯ij = τij 0 01×n and F¯ij is defined by: Fij 0n×n F¯ij = 01×(n−1) 01×n . dx¯¯ =
0n×(n−1)
In
Now consider a Lyapunov function of the following form: N
V¯¯ (x, e) =
βi (x) Vi (x) + eT Me =
i =1
N
βi (x) V¯ i (x, e)
(17)
i=1
where M = M T ∈ ℜn×n and βi (x) , Vi (x) and V¯ i (x) are defined as follows:
βi (x) =
1 0
x ∈ Ωi x ̸∈ Ωi
(18)
Pi qi x P qi , P¯i = i ∗ ri 1 ∗ ri P¯i 0 x¯ P¯ 0 , P¯¯ i = i eT ∗ M e ∗ M
Vi (x) = x¯ T P¯ i x¯ = xT
1
T V¯ i (x) = x¯¯ P¯¯ i x¯¯ = x¯ T
(19)
(20)
where Pi = PiT ∈ ℜn×n , qi ∈ ℜn , ri ∈ ℜ and ∗ denotes symmetry. Given the fact that the system is piecewise affine with respect to the systems states, the Lyapunov function is chosen such that to be piecewise quadratic with respect to the systems states. However, since the observer error vector does not affect the discrete state of the system, a traditional quadratic function is selected as the segment of the Lyapunov function allocated with the observer errors. The Lyapunov function defined in (17) may be utilized to ensure the second moment stability of the stochastic piecewise affine dynamics in (15) on the condition that it remains continuous across the border of each pair of level-1 neighboring cells. The segment of the Lyapunov function in (17) allocated with the observer errors is continuous in nature. Considering (2) and (19), the continuity of the piecewise quadratic segment of (17) may be ensured by forcing the following equality constraints [6]: FijT Pi − Pj Fij = 0 FijT lTij
Pi − Pj lij +
FijT
(21a)
qi − qj = 0
(21b)
Pi − Pj lij + 2(qi − qj ) lij + (ri − rj ) = 0.
T
(21c)
Temporarily assuming that there exists no uncertainty and additive noise in the system, (15) may be simplified as:
˙x¯¯ = A¯ i + B¯ i K¯ i 0n×(n+1)
−B¯ i Ki ¯ x¯ , Ai − Li Ci
x ∈ Ωi
i ∈ I.
(22)
In piecewise affine systems, the equilibrium points of the subspaces not containing the origin do not necessarily have to be located at the origin [6]. Since the continuous state space of the system in this research is not partitioned with respect to the observer error vector, e, the equilibrium points for the observer error dynamics of all the subspaces are designed to be placed at the origin. Thus the deterministic dynamic equation in (22) implies that, for each cell, there exists a local eq equilibrium point for the system, xi , satisfying the following equation [6]: eq (Ai + Bi Ki ) xi + ai + Bi ki = 0,
for i ∈ I .
(23)
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The piecewise quadratic segment of (17) may be utilized to ensure the asymptotic stability of the deterministic system in eq (22) by redefining Vi (x) with respect to the placement of each region’s equilibrium point, xi as: eq T
Vi ( x ) = x − x i
eq
Pi x − xi
+ r˜i
(24)
where r˜i is a scalar constant. Assuming that the piecewise quadratic segment of the Lyapunov function defined in (17), guarantees the asymptotic stability of the system if there were no system uncertainty and additive noise, and comparing (19) and (24), it may be deduced that: eq
qi = −Pi xi .
(25)
As will be illustrated later, the controller gain values in (12) and the observer gain values in (13) may be determined by means of solving an optimization problem constrained by a number of matrix inequalities. To decrease the order of nonlinearity eq in these constraints, xi ’s must be chosen a priori and the manner in which they are selected, directly affects the stability of the closed loop system [6]. In particular, given our objective to stabilize the system on the origin, the following equality constraint must be imposed: eq
xi = 0,
for i ∈ I0 .
(26)
Remark 3. If system (3) without noise (gi (x) ≡ 0, ∀i ∈ I) is free of sliding surfaces, addition of stochastic noise could not add any sliding surface in which the system trajectory remains on indefinitely. If the sliding boundaries are present in the system without noise, the introduction of stochastic noise to the system may result in two possible scenarios. If there is noise present at the direction perpendicular to the sliding boundary, due to the stochastic fluctuations of trajectories, the probability that the system trajectory stays on that sliding boundary for a period of time, is zero. However, if the stochastic perturbations in the system are parallel to the stable sliding boundary, the system trajectory does enter the aforementioned boundary. Should such a phenomena occur, the analysis of the system response after entering the sliding surface is possible in the reduced order state space of the sliding surface. Also as mentioned by Wu et al. [21], the sliding mode analysis in boundaries of subsystems for stochastic hybrid system is an open problem, and each case should be studied individually. In the current study the analysis of the system response is done considering conditions to avoid the sliding modes in the system. Considering (16) and (22), the constraints needed to avoid sliding modes in the closed loop dynamics of (22) are obtained as [6]:
τ¯
T ij
−B¯ i Ki + B¯ j Kj F¯ij s¯ + ¯lij Ai − Aj − Li Ci + Lj Cj Θij Bj Kj − Bi Ki Fij sx + lij
A¯ i − A¯ j + B¯ i K¯ i − B¯ j K¯ j 0
τij 0
=
0n×1
T
Φij
01×n 0n×n
0 0n×1
01×n
Ψij
1 se
= τijT Φij Fij sx + τijT Bj Kj − Bi Ki se + τijT Φij lij + Θij = 0 T where s¯T = sx se , sx ∈ ℜn−1 and se ∈ ℜn are arbitrary vectors and Φij , Θij and Ψij are defined by:
(27)
Φij = Ai − Aj + Bi Ki − Bj Kj Θij = ai − aj + Bi ki − Bj kj Ψij = Ai − Aj − Li Ci + Lj Cj . Since sx and se are arbitrary, it may be deduced from (27) that:
τijT Ai − Aj Fij = 0 τijT Ai − Aj lij + ai − aj + Bi ki − Bj kj = 0 τijT Bi Ki − Bj Kj = 0.
(28a) (28b) (28c)
For the deterministic system in (22), sufficient conditions for sliding mode avoidance are introduced in (28). Given the earlier discussion, enforcing conditions (28) in the observer–controller design will result in the closed loop system (15) without sliding surfaces. Remark 4. The system given in (15) is stochastically second moment stable with bound d > 0 and symmetric weight matrices Qx > 0 and Qe > 0, allocated with the system states and observer errors, respectively; if starting from any initial condition: lim E x (t )T Qx x (t ) + e (t )T Qe e (t ) < d.
t →∞
(29)
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Theorem 1. The uncertain stochastic piecewise affine dynamics in (15) is second moment stabilizable, with bound d > 0 and symmetric weight matrices Qx > 0 and Qe > 0, by implementing the control and observation laws introduced in (12) and (13), if there exists a solution to the following optimization problem: minimize
d
subject to:
P¯ i
T 0 − E¯¯ i Ui E¯¯ i > 0 M
∗ (1, 1)
P¯¯ i
−
εiB ¯ 2
I
(30a)
P¯¯ i
(1, 4)
0
0
A i I
−ε ¯
0
−
εiB ¯ 2
∗
RTi YiT M
Xi Yi Ri
I
(1, 5) 0 0 <0 0 −εiC ¯I
(30b)
>0
(30c)
trace SiT Pi Si + SiT MSi + trace (Xi ) < d
(30d)
and the equality constraints in (21), (23), (25), (26) and (28) hold. Ui ≻ 0, Wi ≻ 0 for i ∈ I are symmetric matrices with nonnegative entries and appropriate dimensions. Xi , for i ∈ I, is a positive definite symmetric matrix with appropriate dimensions and Yi ∈ ℜn×l , for i ∈ I. εiA > 0, εiB > 0 and εiC > 0, for i ∈ I, are positive scalars. ¯I denotes the identity matrix of dimension
(2n + 1). E¯¯ i = E¯ i [0]Ni ×n and Q¯¯ , (1, 1), (1, 4) and (1, 5) are defined by: Qx [0]n×1 [0]n×n ¯ ¯ [0]1×n 0 Q = ∗ Qe T ¯Ai + K¯ iT B¯ Ti P¯i + P¯i A¯ i + B¯ i K¯ i −P¯i B¯ i Ki (1, 1) = −KiT B¯ Ti P¯i ATi M + MAi − CiT YiT − Yi Ci 2 2 T E 0 E 0 + Q¯¯ + E¯¯ W E¯¯ + 2ε A iA¯ + ε C iC¯ i i
i
(1, 4) =
B T T εi K¯ i EiB¯ 0
i
0 εiB KiT EiTB¯
0
0
i
0
0
0 (1, 5) = 0
0 . Yi
Finally the observer gain values of (13) may be obtained from: Li = M −1 Yi .
(31)
Proof. The Lyapunov function defined in (17) is utilized to ensure the second moment stability of the stochastic piecewise affine dynamics in (15) with respect to Lemma 1. Utilizing the definition of x¯¯ = x¯ T
x¯¯
T
P¯ i
∗
0 ¯ x¯ − E¯¯ i x¯¯ M
T
Ui E¯¯ i x¯¯ = x¯ T P¯ i x¯ + eT Me − E¯ i x¯
T
eT
T
, (30a) implies that:
Ui E¯ i x¯ > 0
(32)
given the fact that Ui has non-negative entries and considering (1), it may be deduced that E¯ i x¯
T
Ui E¯ i x¯ > 0 for T
x ∈ Ωi , i ∈ I , thus the Lyapunov function defined in (17) remains positive definite for all x and e. Defining h x¯¯ = x¯¯ Q¯¯ x¯¯ and utilizing Lemma 1 the constraints needed for the second moment stabilization of the closed loop stochastic piecewise affine dynamics in (15) are obtained as:
x¯¯
T
− B¯ i + 1B¯ i Ki ¯ x¯ ∗ Ai − Li Ci − 1Bi Ki ¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki T P¯i 0 T A ¯ + x¯ x¯¯ ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki T + trace SiT Pi Si + SiT MSi + trace RTi LTi MLi Ri ≤ −x¯¯ Q¯¯ x¯¯ + d.
P¯ i
0 M
A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i 1A˜ i − Li 1C¯ i + 1Bi K¯ i
(33)
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Considering (31) and implementing the Schur Complement on (30b) results in:
A¯ i + B¯ i K¯ i 0
−B¯ i Ki Ai − Li Ci
T
P¯ i
0 P¯ + i M ∗
∗
T E2 + Q¯¯ + E¯¯ i Wi E¯¯ i + 2εiA iA¯
0 M
A¯ i + B¯ i K¯ i 0
−B¯ i Ki Ai − Li Ci
0 E2 + εiC iC¯ 0 0
0
0 1 ¯T − T¯i G¯ − i Ti < 0 0
(34)
¯ i are defined as: where T¯i and G
T¯i = P¯¯ i
P¯¯ i
(1, 5)
(1, 4)
B εi ¯ − 2 I ¯i = G
0
0
−εiA ¯I
0
−
0 0
.
εiB ¯ 2
0
I
−εiC ¯I
∗
Considering (11), the following are deduced regarding the left hand side of (34):
T P¯ i 0 P¯ i 0 A¯ i + B¯ i K¯ i −B¯ i Ki −B¯ i Ki LH = + A i − Li C i ∗ M ∗ M 0 A i − Li C i 2 2 1 0 1 ¯ ¯ 0 0 0 0 0 C EiC¯ A EiA¯ ¯ ¯ + C + 2εi + A P i P i + εi T 0 0 0 0 εi εi 0 MLi 0 Li M T T T 2 K¯ E E ¯ K¯ 0 + 2εiB i iB¯ iB i + B P¯¯ i P¯¯ i + Q¯¯ + E¯¯ i Wi E¯¯ i T T 0 Ki EiB¯ EiB¯ Ki εi T A¯ + B¯ i K¯ i −B¯ i Ki P¯ i 0 P¯ 0 A¯ i + B¯ i K¯ i −B¯ i Ki ≥ i + i 0 A i − Li C i ∗ M ∗ M 0 A i − Li C i T T 1 0 1 0 0 1C¯ i 1C¯ i 0 1A¯ i 1A¯ i 0 + 2εiA + A P¯¯ i P¯¯ i + εiC + C 0 0 0 0 εi εi 0 MLi 0 T T T 2 K¯ 1B¯ i 1B¯ i K¯ i 0 + B P¯¯ i P¯¯ i + Q¯¯ + E¯¯ i Wi E¯¯ i . + 2εiB i 0 KiT 1B¯ Ti 1B¯ i Ki εi
A¯ i + B¯ i K¯ i 0
0 LTi M
(35)
Now utilizing Lemma 2 the following matrix inequalities are obtained: 2ε
A i
T 1A¯ i 1A¯ i 0
=ε
≥ εiC
A i
1A¯ i 1A˜ i
1A¯ i 1A˜ i 0
≥
0
−1C¯ i
=
0 0
T
0 0
T 1C¯ i 1C¯ i
2εiB
1 0 + A P¯¯ i P¯¯ i 0 εi
0
−Li 1C¯ i
T
1A¯ i 1A˜ i
P¯¯ i + P¯¯ i
1 0 + C 0 εi
T
0 0
P¯ i
∗ T
0 0
K¯ iT 1B¯ Ti 1B¯ i K¯ i 0
1 0 + A P¯¯ i P¯¯ i 0 εi
1A¯ i 1A˜ i 0 0
0 0 0 MLi
(36a)
0 0
0 P¯ + i ∗ LTi M P¯ i
∗
0 P¯ + i M ∗
0 LTi M 0 MLi 0 M
0
−1C¯ i
0
−Li 1C¯ i
2 0 + B P¯¯ i P¯¯ i KiT 1B¯ Ti 1B¯ i Ki εi
0 0 0 0
(36b)
H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
1B¯ i K¯ i
≥
T
0
−1Bi Ki
0
1B¯ i K¯ i = 1Bi K¯ i
−1B¯ i Ki −1Bi Ki
1B¯ i K¯ i P¯¯ i + P¯¯ i
0
P¯¯ i + P¯¯ i
+
−1Bi Ki
0
T
0
1Bi K¯ i
−1B¯ i Ki
T
0
P¯¯ i + P¯¯ i
161
0
1Bi K¯ i
−1B¯ i Ki . −1Bi Ki
1B¯ i K¯ i 1Bi K¯ i
−1B¯ i Ki
0
(36c)
Considering (36) it is deduced that:
A¯ i + B¯ i K¯ i 0
+ 2εiA + 2εiB
−B¯ i Ki Ai − Li Ci
1A¯ Ti 1A¯ i 0
T
P¯ i
∗
0 P¯ + i M ∗
0 M
A¯ i + B¯ i K¯ i 0
1 0 1C¯ iT 1C¯ i + A P¯¯ i P¯¯ i + εiC 0 0 εi
K¯ iT 1B¯ Ti 1B¯ i K¯ i 0
0 KiT 1BTi 1Bi Ki
¯
¯
+
−B¯ i Ki Ai − Li Ci
1 0 + C 0 εi
0 0
0 MLi
0 0
0
LTi M
T 2 ¯ ¯ P¯ P¯ + Q¯¯ + E¯¯ i Wi E¯¯ i B i i
εi ¯Ai + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki T P¯i 0 ≥ ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki T A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki P¯ i 0 + + Q¯¯ + E¯¯ i Wi E¯¯ i . ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki
(37)
T
Left and right multiplying the last argument in (37) by x¯¯ and x¯¯ , respectively, while considering (34) and (35), results in:
T x¯¯
T
A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki P¯ i 0 ˜ i − Li 1C¯ i + 1Bi K¯ i ∗ M 1 A A − L C − 1 B K i i i i i x¯¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ − Bi + 1Bi Ki Ai + 1Ai + Bi + 1Bi Ki P¯ 0 + Q¯¯ i + i ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki
T T < − E¯¯ i x¯¯ Wi E¯¯ i x¯¯ = − E¯ i x¯ Wi E¯ i x¯
(38)
T
given the fact that Wi has non-negative entries and considering (1), it is deduced that E¯ i x¯ Wi E¯ i x¯ > 0 for x ∈ Ωi , i ∈ I , thus the left hand side of (38) remains negative definite for x ∈ Ωi , i ∈ I . Now considering (31) and implementing the Schur Complement on (30c) results in:
RTi LTi MLi Ri < Xi .
(39)
The inequalities in (38) and (39), alongside (30d), imply that the inequality constraint in (33) is satisfied. Thus the proof is completed. Remark 5. The parameters and matrices to be determined by solving the constrained optimization problem in Theorem 1 include P¯ i , K¯ i , Yi , Xi , Ui ≻ 0, Wi ≻ 0, εiA > 0, εiB > 0, and εiC > 0, for i ∈ I and d and M. It should be noted that, in addition to the constraints mentioned throughout Theorem 1, the positivity of εiA , εiB and εiC , for i ∈ I, and the element-wise positivity of Ui and Wi , for i ∈ I, must be enforced while solving the constrained optimization problem in Theorem 1. Remark 6. It is notable that Theorem 1 only provides sufficient condition for the second moment stabilizability of the closed loop dynamics (15). In other words if the proposed design method in Theorem 1 fails to produce feasible observer–controller gain values one may not comment on the non-observability or non-controllability of the closed loop dynamics of (15). Remark 7. The observer–controller in (12) and (13) is capable of second moment stabilizing the uncertain stochastic piecewise affine system in (9) only if the information regarding the discrete state of the system (i.e. which one of the subspaces is occupied by the system at each given moment in time) are available and the observer–controller gain values may switch accordingly. Since in most practical implementations access to the exact discrete state of the system is not possible due to the presence of measurement noise, one must resort to a discrete state observer to work in conjunction with the proposed hybrid observer–controller in order to stabilize the system, see for example [30,31]. However, providing an exact mathematical proof for the stability of the continuous time observer error dynamics with the discrete state observer is very intricate and to the best of the authors’ knowledge remains as an open problem to this date. It is notable that the emphasis of this paper is on obtaining the stabilizing observer and controller gain values while designing a discrete state observer that guarantees the stability of the system with the PWA observer–controller is left for future researches.
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H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167 Table 1 Parameters of the helicopter model from [32]. Parameter
Value
Unit
Iyy mheli lcgx lcgz Fv M Fv M
0.0283 0.9941 0.0134 0.0289 0.0041 9.81
kg m2 kg m m N m/rad/s m/s2
4. Simulation results In this section, the observer–controller design algorithm proposed in the previous section is implemented on a simplified helicopter pitch dynamics model presented in [32]. The constrained optimization problem arising in Theorem 1 is solved using the solver PENBMI [33] interfaced via the toolbox YALMIP [34]. The state space model of the pitch and pitch rate dynamic equations of the helicopter model are given below:
d
x1 x2
=1
Iyy
x2 −mheli lcgx g cos (x1 ) dt + Sdw −mheli lcgz g sin (x1 ) −FvM x2 + u
(40a)
dz =
x1 dt + Rdv x2
(40b)
where x1 and x2 represent the pitch angle and pitch rate respectively. Iyy is the second moment around y-axis, mheli is the mass of helicopter, lcgx and lcgz are distance from center of mass relative to Joint B (see Fig. 1), Fv M is the pitch damping, and u is the control torque exerted by the main blade of helicopter around y-axis. The values of the parameters are shown in Table 1. It is evident from (40) that the system nonlinearity is only dependent on x1 and may be formulated as: f¯ (x1 ) =
−mheli g Iyy
lcgx cos (x1 ) + lcgz sin (x1 ) .
(41)
In order to find a piecewise linear approximation of the nonlinear dynamics, the state space of the nonlinear system is partitioned into the following subspaces:
9π 2π 2π Ωi = ( x 1 , x 2 ) | − + i ≤ x1 ≤ −π + i , 7
7
7
i = 1, . . . , 7.
(42)
Using a uniform grid of points, the piecewise linear approximation of the system nonlinearity, f¯ ∗ (x1 ), in the subspaces not containing the origin, is calculated such that the sum of the squared error between f¯ (x1 ) and f¯ ∗ (x1 ) is minimized and the approximation remains C 0 . In the subspace containing the origin, the linearization of the nonlinear dynamics at the origin is utilized as f¯ ∗ (x1 ). As mentioned before the uncertainty in the approximated PWA system arises in the form of the difference between f¯ (x1 ) and f¯ ∗ (x1 ). Denoting the squared error between f¯ (x1 ) and f¯ ∗ (x1 ) by g¯ (x1 ) and considering (11a) one may conclude: g¯ (x1 ) = f¯ (x1 ) − f¯ ∗ (x1 )
=
N
2
βi (x) (f (x) − Ai x − ai )T (f (x) − Ai x − ai )
i =1
=
N
βi (x) (1Ai x + 1ai )T (1Ai x + 1ai )
i =1
≤
N
βi (x) x¯ T Ei2A¯ x¯ = g¯ ∗ (x1 )
(43)
i =1
where g¯ ∗ (x1 ) is the piecewise quadratic upper bound of g¯ (x1 ) and βi (x) is the same as defined in (18). Given the positivity of g¯ (x1 ) , g¯ ∗ (x1 ) is easily obtained. The system nonlinearity and its piecewise linear approximation are demonstrated in Fig. 2 alongside g¯ (x1 ) and g¯ ∗ (x1 ). The augmented system matrices of the approximated PWA system are given below:
0 ¯A1 = 6.968 0
1 −0.145 0
0 27.028 , 0
1.712
E12A¯
= ∗
0 0
4.753 0 13.240
(44a)
H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
163
Fig. 1. Pitch model of the helicopter from [32].
Fig. 2. (a) System nonlinearity of simplified helicopter dynamics and its PWA approximation, (b) Squared error between system nonlinearity (g¯ (x)) and its quadratic upper bound (g¯ ∗ (x1 )).
A¯ 2 =
0 −2.128 0
0 ¯A3 = −10.728 0
1 −0.145 0 1 −0.145 0
0 6.617 , 0
E22A¯
0 −4.963 , 0
2.438
= ∗
1.760
E32A¯
0 0
= ∗
4.439 0 8.174
0 0
1.397 0 1.213
(44b)
(44c)
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H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
0 −9.959 0
1 −0.145 0
0 −4.618 , 0
E42A¯
0 ¯A5 = −2.846
1 −0.145 0
0 −7.810 , 0
E52A¯
A¯ 4 =
0
0 ¯A6 = 7.172
1 −0.145 0
0
0 ¯A7 = 11.152
35.336
B¯ i = 0 C¯ i =
1 0
0 1
T
0
0 , 0
0 −30.230 , 0
=
EiB¯ = 0,
0 0 0
0 0
−1.904
∗ 1.552
=
0 0
∗
Ei2C¯ = 0,
2.402
E62A¯
0 0
∗
,
1.822
=
0 −21.298 , 0
1 −0.145 0
0
E72A¯
0.290
= ∗
0 0
(44d)
0 1.592
−3.192
(44e)
0 6.603
(44f)
−0.811 0 2.267
(44g) (44h)
i∈I
i ∈ I.
(44i)
The identity matrix is selected for the weighting matrices, Qx and Qe . For the process noise matrix, Si , and sensor noise matrix, Ri , it is assumed that, 0.6 Si = 0
0 , 0.6
0 Ri = , 0.1
i ∈ I.
(45)
Since there are no uncertainties in the augmented input and output matrices, (30b) may be reduced to:
(1, 1) ∗
P¯¯ i
−εiA ¯I
< 0.
As mentioned before, the local closed loop equilibrium point of each subspace must be selected a priori. Apart from the subspace containing the origin, the equilibrium points for the rest of the subspaces are chosen outside their respective regions so that when the state of the system enters a region not containing the origin, it will eventually leave that region and is stabilized in the region containing the origin. eq
eq
eq
eq
x1 = −x7 = x3 = −x5 =
4.4880 , 0
x2 = −x6 =
0.8976 , 0
x4 =
eq
eq
eq
2.6928 , 0
(46)
0 . 0
Using the system matrices presented in (44) and (45), and the local closed loop equilibrium points assigned in (46), Theorem 1 is utilized to calculate the observer and controller gain values of (12) and (13) and the matrices allocated with the Lyapunov function given in (17). The maximum bound of the second moment stability converges to a value of d = 1.8143 which implies that E xT x + eT e < 1.8143 as time approaches infinity. The rest of the results are presented below:
−2.4690 T ¯K1 = −2.1704 , 9.4307 −3.2076 T K¯ 2 = −1.8225 , 8.6123 −7.5494 T K¯ 3 = −2.3830 , 7.1893 −9.9999 T ¯K4 = −2.5338 , 0.1307 −3.2276 T ¯K5 = −2.2282 , −2.7484
80.2375 L1 = −37.6647
13.6834 4.8802
(47a)
99.4575 −51.4419
3.1721 11.2601
(47b)
169.3088 −92.8010
−5.6440 14.1838
(47c)
6.2453 2.9793
(47d)
L2 =
L3 =
204.1161 L4 = −107.5031
303.6090 316.7910
L5 =
0.3293 0.6848
(47e)
H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
165
Fig. 3. Lyapunov function, (a) Piecewise quadratic segment allocated with system states, (b) Quadratic segment allocated with observer errors.
K¯ 6
K¯ 7
−2.9168 T = −2.6143 , −6.7050 −1.9400 T = −2.5707 , −6.4349
1.0047
P¯ 1 =
79.1474 L6 = −37.2775
13.2070 4.9029
(47f)
77.5300 L7 = −34.3340
19.5708 . 1.6702
(47g)
ε1A = 6.5982
(48a)
ε2A = 4.5979
(48b)
ε3A = 3.9556
(48c)
0.0053 0.0105
∗ 1.3747
P¯ 2 =
0.0072 0.0105
∗ 2.8319
P¯ 3 =
0.0129 0.0105
∗ 4.2439
P¯ 4 =
0.0388 0.0105
∗ 1.6984
P¯ 5 =
1.0156
P¯ 6 =
0.7262
P¯ 7 =
0.0919
∗
(48d)
ε5A = 2.9714
(48e)
0.0072 0.0105
2.7348 0.0194 , −2.8773
ε6A = 3.9209
(48f)
0.0053 0.0105
3.2592 0.0237 , −3.7731
ε7A = 6.4446
(48g)
∗ M =
ε4A = 2.1379
1.5245 0.0116 , −0.8556
∗
0 0 , 0
0.0129 0.0105
∗
−4.5091 −0.0237 , −4.2395 −3.7017 −0.0194 , −2.4784 −2.5419 −0.0116 , −1.9972
0.0995 . 0.1718
(48h)
The level surfaces of the obtained Lyapunov function are presented in Fig. 3 which exhibits its positive definiteness. Thus Theorem 1 shows that the closed loop system with the designed PWA observer–controller is stochastically stable with given upper bound on the second moment.
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H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
Fig. 4. Simulated Trajectories, (a) PWA system with PWA Obs/PWA Ctrl traj. 1, (b) PWA system with PWA Obs/PWA Ctrl traj. 2, (c) Nonlinear system with PWA Obs/PWA Ctrl traj. 1, (d) Nonlinear system with PWA Obs/PWA Ctrl traj. 2, (e) Nonlinear system with EKBF/PWA Ctrl traj. 1, (f) Nonlinear system with EKBF/PWA Ctrl traj. 2, (g) Nonlinear system with UKBF/PWA Ctrl traj. 1, (h) Nonlinear system with UKBF/PWA Ctrl traj. 2.
Table 2 Simulation results.
System
Controller
Observer
E xT x
PWA Nonlinear Nonlinear Nonlinear
PWA PWA PWA PWA
PWA PWA EKBF UKBF
0.7202 1.0757 2.1956 1.6495
E eT e
0.0750 0.1087 0.2650 0.1959
E xT x + eT e
0.7952 1.1844 2.4606 1.8454
The designed PWA observer–controllers are then applied on both the piecewise approximation model and the original nonlinear system for the following different initial conditions:
−3.00 , 1.50 3.00 Initial Condition 2: x0 = , −1.50
Initial Condition 1: x0 =
−2.85 , 1.25 2.85 xˆ 0 = . −1.25
xˆ 0 =
(49)
For comparison purposes, an extended Kalman–Bucy filter (EKBF) and an unscented Kalman–Bucy filter (UKBF), both in combination with the designed PWA controller, are applied on the nonlinear system. The initial conditions for the combined EKBF/PWA controller and the combined UKBF/PWA controller are the same as in (49) and the initial conditional covariance matrix for the EKBF and UKBF is assumed to be P0 = 0.05I. Utilizing the notation introduced in [35] the tuning parameters for the UKBF are set as α = 1, β = 2 and κ = 1. The system trajectories in phase space for 50 s are given in Fig. 4. As can be seen in Fig. 4, the piecewise approximation model and the nonlinear system responses are very similar which indicates that the approximation is close to the nonlinear system and the designed observer–controllers have very similar effects on both systems. As explained in Remark 1, the system is ergodic, thus the time average of xT x + eT e may be utilized to verify
the calculated value of d. The calculated time average of xT x + eT e using both the piecewise affine and nonlinear system trajectories are given in Table 2, both of which comply to the maximum calculated value of d = 1.8143. Other numerical results presented in Table 2 show that the PWA observer–controller has performed more efficiently compared to both the combined EKBF/PWA controller and the combined UKBF/PWA controller.
H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
167
5. Conclusion The problem of minimizing the bound on the variance of the steady response of a class of uncertain piecewise affine systems, subject to system and measurement additive noises, by means of a piecewise affine observer–controller, is formulated as an optimization problem subject to a number of constraints in the form of matrix inequalities. As a practical example, the proposed control scheme is utilized to stabilize the pitch dynamics of a simplified helicopter model. The nonlinear system dynamics are first approximated with an uncertain PWA system where the uncertainties arise in the form of the difference between the actual nonlinear dynamics and the PWA approximation. Simulation results show the efficiency of the control scheme in dealing with the effects of system uncertainties while minimizing the effects of external disturbances on the steady response. The performance of the PWA observer–controller is compared with an extended Kalman–Bucy filter and unscented Kalman–Bucy filter both in combination with the PWA controller. In addition to the fact that the stability of the system is only guaranteed in the case of the PWA observer–controller, it is observed that the PWA observer–controller has performed more efficiently compared to both the combined EKBF/PWA controller and the combined UKBF/PWA controller. Future research includes designing a hybrid discrete state observer to work in conjunction with the proposed PWA observer–controller. References [1] D. Mignone, G. Ferrari-Trecate, M. Morari, Stability and stabilization of piecewise affine and hybrid systems: An LMI approach, in: Proceedings of the 39th IEEE Decision and Control, Vol. 1, IEEE, 2000, pp. 504–509. [2] L. Rodrigues, A. Hassibi, J. How, Output feedback controller synthesis for piecewise-affine systems with multiple equilibria, in: Proceedings of American Control Conference, Vol. 3, IEEE, 2000, pp. 1784–1789. [3] L. Rodrigues, S. Boyd, Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization, Systems Control Lett. 54 (9) (2005) 835–853. [4] A. Hassibi, S. Boyd, Quadratic Stabilization and Control of Piecewise-Linear Systems, Vol. 6, IEEE, 1998, pp. 3659–3664. [5] A. Rantzer, M. Johansson, Piecewise linear quadratic optimal control, IEEE Trans. Automat. Control 45 (4) (2000) 629–637. [6] L. Rodrigues, J. How, Observer-based control of piecewise-affine systems, Internat. J. Control 76 (5) (2003) 459–477. [7] A. Ben Abdallah, M. Hammami, J. Kallel, Robust stability of uncertain piecewise-linear systems: LMI approach, Nonlinear Dynam. 63 (1–2) (2011) 183–192. [8] R. Oliveira, P. Peres, LMI conditions for the existence of polynomially parameter-dependent Lyapunov functions assuring robust stability, in: 44th IEEE Conference on Decision and Control, pp. 1660–1665. [9] J. Xu, L. Xie, Homogeneous polynomial Lyapunov functions for piecewise affine systems, in: Proceedings of the 2005 American Control Conference, pp. 581–586. [10] G. Feng, Controller design and analysis of uncertain piecewise-linear systems, IEEE Trans. Circuits Syst. I 49 (2) (2002) 224–232. [11] H. Zhang, G. Feng, C. Dang, Stability analysis and H∞ control for uncertain stochastic piecewise-linear systems, IET Control Theory Appl. 3 (8) (2009) 1059–1069. [12] M. Bujorianu, J. Lygeros, Toward a general theory of stochastic hybrid systems, Stochastic Hybrid Syst. (2006) 3–30. [13] D. Dimarogonas, K. Kyriakopoulos, Lyapunov-like stability of switched stochastic systems, in: Proceedings of American Control Conference, Vol. 2, IEEE, 2004, pp. 1868–1872. [14] A. Abate, L. Shi, S. Simic, S. Sastry, A stability criterion for stochastic hybrid systems, in: Mathematical theory of Networks and Systems Conference, 2004. [15] J. Xiong, J. Lam, Z. Shu, X. Mao, Stability analysis of continuous-time switched systems with a random switching signal. [16] C. Yuan, J. Lygeros, On the exponential stability of switching diffusion processes, IEEE Trans. Automat. Control 50 (9) (2005) 1422–1426. [17] Y. Dong, J. Sun, Brief paper: On hybrid control of a class of stochastic non-linear Markovian switching systems, Automatica 44 (4) (2008) 990–995. [18] E. Boukas, Static output feedback control for stochastic hybrid systems: LMI approach, Automatica 42 (1) (2006) 183–188. [19] D. De Farias, J. Geromel, J. Do Val, O. Costa, Output feedback control of Markov jump linear systems in continuous-time, IEEE Trans. Automat. Control 45 (5) (2000) 944–949. [20] L. Rodrigues, N. Gollu, Analysis and state feedback control for PWA systems with additive noise, in: Proceedings of American Control Conference, IEEE, 2006, pp. 5438–5443. [21] Z. Wu, M. Cui, P. Shi, H.R. Karimi, Stability of stochastic nonlinear systems with state-dependent switching, IEEE Trans. Automat. Control 58 (8) (2013) 1904–1918. [22] S. Battilotti, Observer design for nonlinear systems with Markov chain, Internat. J. Robust Nonlinear Control 19 (14) (2009) 1603–1631. [23] J.H. Ahrens, H.K. Khalil, High-gain observers in the presence of measurement noise: A switched-gain approach, Automatica 45 (4) (2009) 936–943. [24] J. Le Ny, E. Feron, M. Dahleh, Scheduling Kalman Filters in Continuous Time, IEEE, 2009, pp. 3799–3805. [25] K. Merat, H. Salarieh, A. Alasty, A. Meghdari, Stochastic piecewise affine control with application to pitch control of helicopter, Nonlinear Anal. Hybrid Syst. 15 (2015) 86–97. [26] S. Aberkane, J. Ponsart, M. Rodrigues, D. Sauter, Output feedback control of a class of stochastic hybrid systems, Automatica 44 (5) (2008) 1325–1332. [27] F. Xi, G. Yin, The strong feller property of switching jump-diffusion processes, Statist. Probab. Lett. 83 (3) (2013) 761–767. [28] M. Zakai, A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise, SIAM J. Control 7 (1969) 390. [29] I.R. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems Control Lett. 8 (4) (1987) 351–357. [30] C.E. Seah, I. Hwang, State estimation for stochastic linear hybrid systems with continuous-state-dependent transitions: an IMM approach, IEEE Trans. Aerosp. Electron. Syst. 45 (1) (2009) 376–392. [31] C.E. Seah, I. Hwang, Algorithm for performance analysis of the IMM algorithm, IEEE Trans. Aerosp. Electron. Syst. 47 (2) (2011) 1114–1124. [32] R. Endreß, L. Rodrigues, Modeling and control of two degrees of freedom quanser helicopter, Tech. Rep., Concordia University, 2008. [33] M. Kočvara, M. Stingl, Pennon: A code for convex nonlinear and semidefinite programming, Optim. Methods Softw. 18 (3) (2003) 317–333. [34] J. Lofberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in: IEEE International Symposium on Computer Aided Control Systems Design, IEEE, 2004, pp. 284–289. [35] S. Sarkka, On unscented Kalman filtering for state estimation of continuous-time nonlinear systems, IEEE Trans. Automat. Control 52 (9) (2007) 1631–1641.
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Observer based minimum variance control of uncertain piecewise affine systems subject to additive noise Hamidreza Razavi, Kaveh Merat, Hassan Salarieh ∗ , Aria Alasty, Ali Meghdari Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Azadi Ave, Tehran, Iran
article
info
Keywords: Hybrid observer–controller Non-vanishing additive noise Stochastic hybrid systems Piecewise quadratic Lyapunov function
abstract In this paper, a class of uncertain piecewise affine (PWA) systems, subject to system and measurement additive noises is studied. The additive noise signals considered here do not vanish at the equilibrium and the uncertainties are norm bounded. The problem of minimizing the bound on the variance of the steady response of uncertain PWA systems, by means of a hybrid observer–controller, is formulated as an optimization problem subject to a number of constraints in the form of matrix inequalities. The derived constraints are obtained by considering a piecewise quadratic Lyapunov function in combination with the general stability conditions regarding the existence of an upper stochastic bound on the steady state variance for a class of stochastic hybrid systems (SHS). Then the uncertain PWA approximation of a practical system with nonlinear dynamics is presented considering system and measurement noises. The uncertainties arise in the form of the difference between the actual nonlinear dynamics and the PWA approximation. Utilizing the introduced methods, a hybrid observer–controller is designed and implemented on the nonlinear system to demonstrate the effectiveness of the proposed controller design procedure. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Piecewise affine (PWA) systems are a powerful modeling tool for the control of nonlinear systems. These systems constitute a class of hybrid systems and are often encountered in control systems where piecewise-linear components such as dead-zone, saturation, relays and hysteresis are present. PWA systems may also be used to approximate the dynamics of other nonlinear systems. The stability analysis of deterministic PWA systems based on the multiple Lyapunov function method was studied by many authors [1–3]. Also, the state feedback control of PWA systems has attracted many researches [4,5,3] which consider deterministic systems controlled by continuous-time controllers. The simultaneous observer and controller design for deterministic PWA systems was studied by Rodrigues and How [6]. The PWA systems considered here however, are more generalized; as they include both norm bounded uncertainty terms and additive noise terms in both the process and measurement equations. Many practical systems, including PWA systems, are subject to uncertainties due to modeling error and external disturbances. In the case of using a PWA system to model the behavior of a nonlinear system, the PWA system will be subjected to more approximation error, thus increasing the uncertainty in the model. Several approaches are proposed for the controller design of uncertain PWA systems. Ben Abdallah et al. [7] assumed that the uncertain system parameters belong to a convex polyhedral and utilized the results obtained by Oliveira and Peres [8] and Xu and Xie [9]. It was proved that obtaining a solution that satisfies the related matrix inequality constraints at the vertices of the mentioned convex polyhedral will result
∗
Corresponding author. Tel.: +98 21 6616 5538; fax: +98 21 6600 0021. E-mail address: [email protected] (H. Salarieh).
http://dx.doi.org/10.1016/j.nahs.2015.09.002 1751-570X/© 2015 Elsevier Ltd. All rights reserved.
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in obtaining the desired controller gain values and piecewise quadratic Lyapunov function (PQLF) ensuring the closed loop system stability. However, as the number of system states, partitions and uncertainty polyhedral vertices increase, obtaining a solution by this method becomes exponentially difficult. Feng [10] proposed a method in which only the upper bounds of the system uncertainties are included in the inequalities. Zhang et al. [11] modeled the system uncertainties similar to Feng [10]. The authors further generalized the model by including a one dimensional stochastic noise term that vanishes at the steady state equilibrium point. The piecewise affine controller design problem was then formulated as an optimization problem subject to a number of matrix inequality constraints in order to achieve stochastic stability in the mean square. However this design has limited applicability as in practice the noise terms do not usually vanish at the steady state equilibrium point and, as will be discussed later, achieving stochastic stability in the mean square is impossible. The inclusion of the additive noise terms in the PWA systems studied here makes them a subset of the more generalized stochastic hybrid systems (SHS). The SHSs comprise a vast range of systems where some randomness is introduced to a hybrid model, with both discrete and continuous states. The randomness in the SHS could be considered in both the evolution of the discrete state and dynamics of the continuous states. Bujorianu and Lygeros [12] proposed a general structure for SHS where the continuous evolution is defined by a stochastic diffusion process and the governing relation for the discrete state is probabilistic. In this framework, both discrete and continuous dynamics could be interconnected, in other words both of them could be dependent on the discrete and continuous states. Given that the design of controller and observer for the general SHS is very challenging in its general format, many methods have been studied that consider a special class of general SHS for stability analysis and controller design. The Lyapunov based stability theorems for different classes of stochastic hybrid systems have been widely investigated. Dimarogonas and Kyriakopoulos [13] gave stochastic stability theorems for a class of SHS with arbitrary change of discrete state. The proposed theorems are based on conditions that guarantee the stability of the steady state equilibrium point with probability one regardless of the discrete state dynamics and with noise that vanishes at the steady state equilibrium point. It is notable that discarding the discrete dynamics can simplify the conditions but may result in very conservative sufficient conditions. The asymptotic stability condition for SHS with a Markov process as the discrete evolution dynamics is investigated by Abate et al. [14] where it is assumed that the jumps have probabilistic dwell-time in each discrete state. More recently Xiong et al. [15] derived the stability condition, in the form of Linear Matrix Inequalities (LMI), for switching linear systems. For a class of stochastic hybrid nonlinear systems with Markov jumps, the almost sure exponential stability theorem is given by Yuan and Lygeros [16]. Again there is a condition that requires the noise term to disappear at the steady state equilibrium point. It is remarkable that exponential and asymptotic stability, or stability with probability one, are not achievable in the presence of noise that remains non-zero at the steady state equilibrium point. Moving on to the controller design for SHS, Dong and Sun [17] have introduced the impulsive output feedback control for a class of nonlinear SHS with Markov jumps. Boukas [18] addressed the static output feedback control for an uncertain singular linear stochastic system with Markov switches, where there are norm bounded uncertainties and the noise signals vanish at the equilibrium. Another output feedback control design via LMI for continuous-time Markovian jump linear systems is proposed by De Farias et al. [19], which again guarantees the exponential stochastic mean square stability. Most of the literature for SHS stability and control design have not considered state dependent jumps and additive noises that do not vanish in the equilibrium point. Both of the aforementioned properties are required for the stochastic uncertain PWA systems considered here. In the case addressed in this article, asymptotic stability for mean square variance is impossible and only a bound on the second stochastic moment could be obtained. Similarly Rodrigues and Gollu [20] studied the control of a piecewise stochastic system with additive noise, but the common Lyapunov function method is applied and it is assumed that the drift and noise terms are continuous at the boundaries between subsystems and also no upper bound on the variance is given. One of the other close studies in the stability of stochastic hybrid systems with state-dependent switching is given by Wu et al. [21]. The stability theorem approach given is based on the generalization of Barbalat lemma to the stochastic hybrid systems. But still the given stability theorems consider vanishing noise at equilibrium which results in mean square stability. The state estimation for SHS has also been the topic of many researches. Battilotti [22] proposed a novel observer for SHS with Markov jumps satisfying some dwell-time. This observer guarantees that the state estimation error of the switching dynamics converges asymptotically to zero with probability one. Ahrens and Khalil [23] developed high-gain observers for a class of nonlinear systems with measurement noise. So there is no discrete state in the main system but the observer jumps between two gain values. Le Ny et al. [24] proposed scheduling Kalman filters for the observation of a class of stochastic linear systems where the Kalman gains are obtained via LMI. Most of the studies in observer design, consider Markov jumps in order to simplify the stability analysis. In this article however, state dependent jumps are considered. To the best of the authors knowledge however, the problem of stabilizing an uncertain stochastic PWA system by means of a piecewise affine observer–controller has never been addressed in the literature. Thus in this article, a systematic method is proposed for simultaneous design of observer and controller for uncertain stochastic PWA systems. In this system, the noises are additive and non-vanishing at the steady state equilibrium point, so the design procedure gives an upper bound on the combined estimation error and controller error variances. On the other hand, as stated before, the uncertainties in the considered PWA system may arise when the PWA system is utilized to model the behavior of a nonlinear system. If a control law for the nonlinear system is to be designed based
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on its approximated PWA system, the upper bound of the uncertainties in the PWA system should be known. Accuracy in the calculation of these upper bounds is crucial as they directly affect the stability of the closed loop system. However, the method of obtaining these upper bounds is seldom addressed in the literature. Thus another contribution of this article is to propose a systematic method to calculate the upper bounds of the uncertainties in the PWA system. The rest of the paper is organized as follows. In the next section, some preliminary definitions and properties are given about a general class of SHS alongside the conditions of their stability. Then in the following section, the design procedure for the combined controller and observer for uncertain stochastic PWA systems is introduced. Next, the proposed design method is implemented on a simplified nonlinear helicopter model to show its performance. The conclusions are drawn in the final section. 2. Preliminaries and hybrid stochastic stability The continuous state space of hybrid systems studied here is partitioned into a number of closed (possibly unbounded) polyhedral cells Ωi , i ∈ I = {1, 2, . . . , N }, such that ∪i∈I Ωi = ℜn . The index set of the cells containing the steady state equilibrium point (Here considered to be located at the origin) is denoted by I0 ⊆ I while the index set of the cells not containing the origin is denoted by I1 ⊆ I. Denoting the state vector of the system by x ∈ ℜn , the augmented state vector is defined by x¯ = xT defined by:
T
1 . Now given the fact that each cell is a polyhedral, the ith cell Ωi , enclosed by Ni facets, may be
Ωi = x|Ei x + ei = E¯ i x¯ ≥ 0 (1) where E¯ i = Ei ei , Ei ∈ ℜNi ×n is a constant matrix, each of whose rows is vector perpendicular to one of the Ni facets enclosing Ωi and ei ∈ ℜNi . Any two cells sharing a common facet are called level-1 neighboring cells. It is assumed that over the common boundary between two adjacent cells, the state of the system follows the dynamic equations of the cell with the lesser index number. A parametric description of the common facet between two level-1 neighboring cells, Ωi and Ωj , may be obtained as [6]:
Ωi ∩ Ωj ⊆ x|x = Fij s + lij , s ∈ ℜn−1 = x|τijT x − lij = 0
(2)
where Fij ∈ ℜn×(n−1) is a constant full rank matrix whose columns comprise a base mapping for the common facet between Ωi and Ωj . lij ∈ ℜn may be chosen as any arbitrary constant point on the common facet and τij ∈ ℜn represents the vector perpendicular to the common facet. A family of stochastic hybrid systems in Ito form, with state dependent jumps, is defined as: dx = fi (x) dt + gi (x) dw,
x ∈ Ωi ,
i∈I
(3)
where x ∈ ℜ is the state variable vector and w (t ) is a q-dimensional normalized Wiener process satisfying E [dw ] = 0 and E (dw) (dw)T = Iq dt with Iq as the q- dimensional identity matrix. fi (x) : ℜn → ℜn and gi (x) : ℜn → ℜn×q for ∀i ∈ I are bounded and Lipschitz continuous to ensure the existence and uniqueness of the corresponding solution process in Ωi . n
Definition 1 ([25]). The system given in (3) is stochastically second moment stable with bound d > 0 and symmetric weight matrix Q > 0 if starting from any initial condition we have: lim E x (t )T Qx (t ) < d.
(4)
t →∞
It is notable that this definition is different from stochastic stability in the mean-square defined in [26,11] as it does not require the variances to tend to zero in time which is only possible when there are multiplicative noises in the system. Definition 2. For each stochastic subsystem in (3) the local infinitesimal generator acting on a twice continuously differentiable scalar function Y ∈ C 2 : ℜn → ℜ is defined by: Li Y (x) =
1 ∂ 2 Y ( x) ∂ Y ( x) fi (x) + trace giT (x) g x . ( ) i ∂x 2 ∂ x2
(5)
Lemma 1 ([25]). Consider the hybrid stochastic system in (3). Let there exist twice continuously differentiable functions Vi : Ωi → ℜ+ (Multi-Lyapunov functions), constant d > 0, and scalar semi-positive (non-negative) function h (x) such that: Vi (x) = Vj (x) ,
x ∈ Ωi ∩ Ωj
Li Vi (x) ≤ −h (x) + d,
x ∈ Ωi ,
(6a) i ∈ I.
(6b)
Then we have: lim E [h (x)] ≤ d.
t →∞
(7)
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Remark 1 ([25]). A system is called Feller process if for all of its solution x (t ) and all bounded and continuous functions f (x), we have continuity for E [f (x)]. It should be noted that with Lipschitz continuity of fi and gi , one can prove the Feller property for the system in (3) using an approach like that of [27]. As mentioned in [28], if the conditions of Lemma 1 are met and the system is a Feller process, then there exists an invariant probability measure for the system. So we may conclude that the system is ergodic and it has steady distribution. Remark 2. In the current study the symbol > (≥) is used to denote matrix inequalities while the symbol ≻ is used to denote element-wise inequalities. Lemma 2 ([29]). For arbitrary matrices X , Y and scalar number ε > 0 we have:
ε X T X + ε −1 Y T Y ≥ X T Y + Y T X .
(8)
3. Observer–controller design for stochastic uncertain piecewise affine systems As a special case for (3), the following Ito diffusion formula may now be introduced to illustrate the dynamic equations of an uncertain stochastic piecewise affine system in each cell: dx = ((Ai + 1Ai ) x + (Bi + 1Bi ) u + ai + 1ai ) dt + Si dw, dz = ((Ci + 1Ci ) x + ci + 1ci ) dt + Ri dv,
x ∈ Ωi
x ∈ Ωi
i∈I
i∈I
(9a) (9b)
where Ai ∈ ℜn×n , ai ∈ ℜn , Bi ∈ ℜn×m , Ci ∈ ℜl×m and ci ∈ ℜl are the state matrix, system affine term, input matrix, output matrix and output affine term of the ith nominal local stochastic linear model, respectively. u ∈ ℜm is the system input vectorand Si dw defines the system additive white noise where w (t ) is a q-dimensional normalized Wiener process satisfying E (dw) (dw)T = Iq dt with Iq as the q-dimensional identity matrix. dz ∈ ℜl is the system output vector and is perturbed by the measurement additive white noise, Ri dv . v (t ) is a p-dimensional normalized Wiener process satisfying E [dv ] = 0 and E (dv) (dv)T = Ip dt with Ip as the p-dimensional identity matrix and Ri ∈ ℜn×p . 1Ai , 1ai , 1Bi , 1Ci and 1ci denote the uncertainty terms in Ai , ai , Bi , Ci and ci , respectively; and are of similar dimensions to that of their corresponding nominal values. It is assumed that w (t ) and v (t ) are uncorrelated. Utilizing the augmented state vector x¯ = xT
T
1 , (12) may be rewritten in the following form: A¯ i + 1A¯ i x¯ + B¯ i + 1B¯ i u dt + S¯i dw,
x ∈ Ωi i ∈ I ¯ ¯ dz = Ci + 1Ci x¯ dt + Ri dv, x ∈ Ωi i ∈ I T T where B¯ i = BTi 0 , C¯ i = Ci ci , S¯i = SiT 0 and A¯ i is defined by: A i ai A¯ i = dx¯ =
0
(10a) (10b)
0
and the uncertainty matrices 1A¯ i , 1B¯ i and 1C¯ i are augmented similar to A¯ i , B¯ i and C¯ i , respectively. Assumption 1. In order to present stability results, it is assumed that the upper bounds of the system uncertainties are known a priori, that is,
T 1ai 1Ai T 1ci 1Ci
1Ai 1Ci
1BTi 1Bi
≤
2 1ai ≤ E¯ iA 1ci ≤ E¯ iC2
(11a) (11b)
T EiB EiB
(11c)
2 where EiB is of the same dimensions as 1Bi and E¯ iA ∈ ℜ(n+1)×(n+1) and E¯ iC2 ∈ ℜ(n+1)×(n+1) are symmetric matrices. It is notable that the definition of the uncertainty upper bound matrices, for the system and output matrices, introduced in (11a) and (11b) do not require the uncertainty upper matrices to be decomposable in the format introduced by Feng in [10] and is thus slightly more general.
The purpose of this study is the second moment stabilizing of the system in (9) by means of a piecewise affine observer–controller of the following form: u = Ki xˆ + ki ,
x ∈ Ωi
i∈I
(12)
where Ki ∈ ℜ , ki ∈ ℜ are the constant gain matrix and offset term of the controller assigned to the ith nominal local stochastic linear model, respectively. xˆ ∈ ℜn is the estimation of the state vector x and satisfies the following Ito diffusion formula: m×n
m
dxˆ = Ai xˆ + Bi u + ai dt + Li dz − Ci xˆ + ci dt ,
x ∈ Ωi ,
i∈I
(13)
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where Li ∈ ℜn×l is the observer gain matrix of the ith nominal local stochastic linear model. Defining e = x − xˆ as the observer error and utilizing the augmented state vector x¯ = following form: u = K¯ i x¯ − Ki e, where K¯ i = Ki
x ∈ Ωi ,
xT
T
1 , the control law in (12) may be rewritten in the (14)
i∈I
ki . Now combining (10), (13) and (14) results in:
A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i 1A˜ i − Li 1C¯ i + 1Bi K¯ i
− B¯ i + 1B¯ i Ki ¯ S¯ 0 x ∈ Ωi , i ∈ I x¯ dt + i dw, ¯ (15) S i −L i R i Ai − Li Ci − 1Bi Ki T T ¯ = w T v T is an augmented (q + p)-dimensional normalized Wiener where 1A˜ = 1Ai 1ai , x¯¯ = x¯ T eT and w T T process. Given the augmented state vector x¯¯ = x¯ eT , a parametric description of the common facet between two level-1 neighboring cells, Ωi and Ωj may be obtained as: (16) Ωi ∩ Ωj ⊆ x¯¯ |x¯¯ = F¯ij s¯ + ¯lij , s¯ ∈ ℜ2n−1 = x¯¯ |τ¯ijT x¯¯ − ¯lij = 0 T T T T where ¯lij = lij 1 01×n , τ¯ij = τij 0 01×n and F¯ij is defined by: Fij 0n×n F¯ij = 01×(n−1) 01×n . dx¯¯ =
0n×(n−1)
In
Now consider a Lyapunov function of the following form: N
V¯¯ (x, e) =
βi (x) Vi (x) + eT Me =
i =1
N
βi (x) V¯ i (x, e)
(17)
i=1
where M = M T ∈ ℜn×n and βi (x) , Vi (x) and V¯ i (x) are defined as follows:
βi (x) =
1 0
x ∈ Ωi x ̸∈ Ωi
(18)
Pi qi x P qi , P¯i = i ∗ ri 1 ∗ ri P¯i 0 x¯ P¯ 0 , P¯¯ i = i eT ∗ M e ∗ M
Vi (x) = x¯ T P¯ i x¯ = xT
1
T V¯ i (x) = x¯¯ P¯¯ i x¯¯ = x¯ T
(19)
(20)
where Pi = PiT ∈ ℜn×n , qi ∈ ℜn , ri ∈ ℜ and ∗ denotes symmetry. Given the fact that the system is piecewise affine with respect to the systems states, the Lyapunov function is chosen such that to be piecewise quadratic with respect to the systems states. However, since the observer error vector does not affect the discrete state of the system, a traditional quadratic function is selected as the segment of the Lyapunov function allocated with the observer errors. The Lyapunov function defined in (17) may be utilized to ensure the second moment stability of the stochastic piecewise affine dynamics in (15) on the condition that it remains continuous across the border of each pair of level-1 neighboring cells. The segment of the Lyapunov function in (17) allocated with the observer errors is continuous in nature. Considering (2) and (19), the continuity of the piecewise quadratic segment of (17) may be ensured by forcing the following equality constraints [6]: FijT Pi − Pj Fij = 0 FijT lTij
Pi − Pj lij +
FijT
(21a)
qi − qj = 0
(21b)
Pi − Pj lij + 2(qi − qj ) lij + (ri − rj ) = 0.
T
(21c)
Temporarily assuming that there exists no uncertainty and additive noise in the system, (15) may be simplified as:
˙x¯¯ = A¯ i + B¯ i K¯ i 0n×(n+1)
−B¯ i Ki ¯ x¯ , Ai − Li Ci
x ∈ Ωi
i ∈ I.
(22)
In piecewise affine systems, the equilibrium points of the subspaces not containing the origin do not necessarily have to be located at the origin [6]. Since the continuous state space of the system in this research is not partitioned with respect to the observer error vector, e, the equilibrium points for the observer error dynamics of all the subspaces are designed to be placed at the origin. Thus the deterministic dynamic equation in (22) implies that, for each cell, there exists a local eq equilibrium point for the system, xi , satisfying the following equation [6]: eq (Ai + Bi Ki ) xi + ai + Bi ki = 0,
for i ∈ I .
(23)
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The piecewise quadratic segment of (17) may be utilized to ensure the asymptotic stability of the deterministic system in eq (22) by redefining Vi (x) with respect to the placement of each region’s equilibrium point, xi as: eq T
Vi ( x ) = x − x i
eq
Pi x − xi
+ r˜i
(24)
where r˜i is a scalar constant. Assuming that the piecewise quadratic segment of the Lyapunov function defined in (17), guarantees the asymptotic stability of the system if there were no system uncertainty and additive noise, and comparing (19) and (24), it may be deduced that: eq
qi = −Pi xi .
(25)
As will be illustrated later, the controller gain values in (12) and the observer gain values in (13) may be determined by means of solving an optimization problem constrained by a number of matrix inequalities. To decrease the order of nonlinearity eq in these constraints, xi ’s must be chosen a priori and the manner in which they are selected, directly affects the stability of the closed loop system [6]. In particular, given our objective to stabilize the system on the origin, the following equality constraint must be imposed: eq
xi = 0,
for i ∈ I0 .
(26)
Remark 3. If system (3) without noise (gi (x) ≡ 0, ∀i ∈ I) is free of sliding surfaces, addition of stochastic noise could not add any sliding surface in which the system trajectory remains on indefinitely. If the sliding boundaries are present in the system without noise, the introduction of stochastic noise to the system may result in two possible scenarios. If there is noise present at the direction perpendicular to the sliding boundary, due to the stochastic fluctuations of trajectories, the probability that the system trajectory stays on that sliding boundary for a period of time, is zero. However, if the stochastic perturbations in the system are parallel to the stable sliding boundary, the system trajectory does enter the aforementioned boundary. Should such a phenomena occur, the analysis of the system response after entering the sliding surface is possible in the reduced order state space of the sliding surface. Also as mentioned by Wu et al. [21], the sliding mode analysis in boundaries of subsystems for stochastic hybrid system is an open problem, and each case should be studied individually. In the current study the analysis of the system response is done considering conditions to avoid the sliding modes in the system. Considering (16) and (22), the constraints needed to avoid sliding modes in the closed loop dynamics of (22) are obtained as [6]:
τ¯
T ij
−B¯ i Ki + B¯ j Kj F¯ij s¯ + ¯lij Ai − Aj − Li Ci + Lj Cj Θij Bj Kj − Bi Ki Fij sx + lij
A¯ i − A¯ j + B¯ i K¯ i − B¯ j K¯ j 0
τij 0
=
0n×1
T
Φij
01×n 0n×n
0 0n×1
01×n
Ψij
1 se
= τijT Φij Fij sx + τijT Bj Kj − Bi Ki se + τijT Φij lij + Θij = 0 T where s¯T = sx se , sx ∈ ℜn−1 and se ∈ ℜn are arbitrary vectors and Φij , Θij and Ψij are defined by:
(27)
Φij = Ai − Aj + Bi Ki − Bj Kj Θij = ai − aj + Bi ki − Bj kj Ψij = Ai − Aj − Li Ci + Lj Cj . Since sx and se are arbitrary, it may be deduced from (27) that:
τijT Ai − Aj Fij = 0 τijT Ai − Aj lij + ai − aj + Bi ki − Bj kj = 0 τijT Bi Ki − Bj Kj = 0.
(28a) (28b) (28c)
For the deterministic system in (22), sufficient conditions for sliding mode avoidance are introduced in (28). Given the earlier discussion, enforcing conditions (28) in the observer–controller design will result in the closed loop system (15) without sliding surfaces. Remark 4. The system given in (15) is stochastically second moment stable with bound d > 0 and symmetric weight matrices Qx > 0 and Qe > 0, allocated with the system states and observer errors, respectively; if starting from any initial condition: lim E x (t )T Qx x (t ) + e (t )T Qe e (t ) < d.
t →∞
(29)
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Theorem 1. The uncertain stochastic piecewise affine dynamics in (15) is second moment stabilizable, with bound d > 0 and symmetric weight matrices Qx > 0 and Qe > 0, by implementing the control and observation laws introduced in (12) and (13), if there exists a solution to the following optimization problem: minimize
d
subject to:
P¯ i
T 0 − E¯¯ i Ui E¯¯ i > 0 M
∗ (1, 1)
P¯¯ i
−
εiB ¯ 2
I
(30a)
P¯¯ i
(1, 4)
0
0
A i I
−ε ¯
0
−
εiB ¯ 2
∗
RTi YiT M
Xi Yi Ri
I
(1, 5) 0 0 <0 0 −εiC ¯I
(30b)
>0
(30c)
trace SiT Pi Si + SiT MSi + trace (Xi ) < d
(30d)
and the equality constraints in (21), (23), (25), (26) and (28) hold. Ui ≻ 0, Wi ≻ 0 for i ∈ I are symmetric matrices with nonnegative entries and appropriate dimensions. Xi , for i ∈ I, is a positive definite symmetric matrix with appropriate dimensions and Yi ∈ ℜn×l , for i ∈ I. εiA > 0, εiB > 0 and εiC > 0, for i ∈ I, are positive scalars. ¯I denotes the identity matrix of dimension
(2n + 1). E¯¯ i = E¯ i [0]Ni ×n and Q¯¯ , (1, 1), (1, 4) and (1, 5) are defined by: Qx [0]n×1 [0]n×n ¯ ¯ [0]1×n 0 Q = ∗ Qe T ¯Ai + K¯ iT B¯ Ti P¯i + P¯i A¯ i + B¯ i K¯ i −P¯i B¯ i Ki (1, 1) = −KiT B¯ Ti P¯i ATi M + MAi − CiT YiT − Yi Ci 2 2 T E 0 E 0 + Q¯¯ + E¯¯ W E¯¯ + 2ε A iA¯ + ε C iC¯ i i
i
(1, 4) =
B T T εi K¯ i EiB¯ 0
i
0 εiB KiT EiTB¯
0
0
i
0
0
0 (1, 5) = 0
0 . Yi
Finally the observer gain values of (13) may be obtained from: Li = M −1 Yi .
(31)
Proof. The Lyapunov function defined in (17) is utilized to ensure the second moment stability of the stochastic piecewise affine dynamics in (15) with respect to Lemma 1. Utilizing the definition of x¯¯ = x¯ T
x¯¯
T
P¯ i
∗
0 ¯ x¯ − E¯¯ i x¯¯ M
T
Ui E¯¯ i x¯¯ = x¯ T P¯ i x¯ + eT Me − E¯ i x¯
T
eT
T
, (30a) implies that:
Ui E¯ i x¯ > 0
(32)
given the fact that Ui has non-negative entries and considering (1), it may be deduced that E¯ i x¯
T
Ui E¯ i x¯ > 0 for T
x ∈ Ωi , i ∈ I , thus the Lyapunov function defined in (17) remains positive definite for all x and e. Defining h x¯¯ = x¯¯ Q¯¯ x¯¯ and utilizing Lemma 1 the constraints needed for the second moment stabilization of the closed loop stochastic piecewise affine dynamics in (15) are obtained as:
x¯¯
T
− B¯ i + 1B¯ i Ki ¯ x¯ ∗ Ai − Li Ci − 1Bi Ki ¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki T P¯i 0 T A ¯ + x¯ x¯¯ ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki T + trace SiT Pi Si + SiT MSi + trace RTi LTi MLi Ri ≤ −x¯¯ Q¯¯ x¯¯ + d.
P¯ i
0 M
A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i 1A˜ i − Li 1C¯ i + 1Bi K¯ i
(33)
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Considering (31) and implementing the Schur Complement on (30b) results in:
A¯ i + B¯ i K¯ i 0
−B¯ i Ki Ai − Li Ci
T
P¯ i
0 P¯ + i M ∗
∗
T E2 + Q¯¯ + E¯¯ i Wi E¯¯ i + 2εiA iA¯
0 M
A¯ i + B¯ i K¯ i 0
−B¯ i Ki Ai − Li Ci
0 E2 + εiC iC¯ 0 0
0
0 1 ¯T − T¯i G¯ − i Ti < 0 0
(34)
¯ i are defined as: where T¯i and G
T¯i = P¯¯ i
P¯¯ i
(1, 5)
(1, 4)
B εi ¯ − 2 I ¯i = G
0
0
−εiA ¯I
0
−
0 0
.
εiB ¯ 2
0
I
−εiC ¯I
∗
Considering (11), the following are deduced regarding the left hand side of (34):
T P¯ i 0 P¯ i 0 A¯ i + B¯ i K¯ i −B¯ i Ki −B¯ i Ki LH = + A i − Li C i ∗ M ∗ M 0 A i − Li C i 2 2 1 0 1 ¯ ¯ 0 0 0 0 0 C EiC¯ A EiA¯ ¯ ¯ + C + 2εi + A P i P i + εi T 0 0 0 0 εi εi 0 MLi 0 Li M T T T 2 K¯ E E ¯ K¯ 0 + 2εiB i iB¯ iB i + B P¯¯ i P¯¯ i + Q¯¯ + E¯¯ i Wi E¯¯ i T T 0 Ki EiB¯ EiB¯ Ki εi T A¯ + B¯ i K¯ i −B¯ i Ki P¯ i 0 P¯ 0 A¯ i + B¯ i K¯ i −B¯ i Ki ≥ i + i 0 A i − Li C i ∗ M ∗ M 0 A i − Li C i T T 1 0 1 0 0 1C¯ i 1C¯ i 0 1A¯ i 1A¯ i 0 + 2εiA + A P¯¯ i P¯¯ i + εiC + C 0 0 0 0 εi εi 0 MLi 0 T T T 2 K¯ 1B¯ i 1B¯ i K¯ i 0 + B P¯¯ i P¯¯ i + Q¯¯ + E¯¯ i Wi E¯¯ i . + 2εiB i 0 KiT 1B¯ Ti 1B¯ i Ki εi
A¯ i + B¯ i K¯ i 0
0 LTi M
(35)
Now utilizing Lemma 2 the following matrix inequalities are obtained: 2ε
A i
T 1A¯ i 1A¯ i 0
=ε
≥ εiC
A i
1A¯ i 1A˜ i
1A¯ i 1A˜ i 0
≥
0
−1C¯ i
=
0 0
T
0 0
T 1C¯ i 1C¯ i
2εiB
1 0 + A P¯¯ i P¯¯ i 0 εi
0
−Li 1C¯ i
T
1A¯ i 1A˜ i
P¯¯ i + P¯¯ i
1 0 + C 0 εi
T
0 0
P¯ i
∗ T
0 0
K¯ iT 1B¯ Ti 1B¯ i K¯ i 0
1 0 + A P¯¯ i P¯¯ i 0 εi
1A¯ i 1A˜ i 0 0
0 0 0 MLi
(36a)
0 0
0 P¯ + i ∗ LTi M P¯ i
∗
0 P¯ + i M ∗
0 LTi M 0 MLi 0 M
0
−1C¯ i
0
−Li 1C¯ i
2 0 + B P¯¯ i P¯¯ i KiT 1B¯ Ti 1B¯ i Ki εi
0 0 0 0
(36b)
H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
1B¯ i K¯ i
≥
T
0
−1Bi Ki
0
1B¯ i K¯ i = 1Bi K¯ i
−1B¯ i Ki −1Bi Ki
1B¯ i K¯ i P¯¯ i + P¯¯ i
0
P¯¯ i + P¯¯ i
+
−1Bi Ki
0
T
0
1Bi K¯ i
−1B¯ i Ki
T
0
P¯¯ i + P¯¯ i
161
0
1Bi K¯ i
−1B¯ i Ki . −1Bi Ki
1B¯ i K¯ i 1Bi K¯ i
−1B¯ i Ki
0
(36c)
Considering (36) it is deduced that:
A¯ i + B¯ i K¯ i 0
+ 2εiA + 2εiB
−B¯ i Ki Ai − Li Ci
1A¯ Ti 1A¯ i 0
T
P¯ i
∗
0 P¯ + i M ∗
0 M
A¯ i + B¯ i K¯ i 0
1 0 1C¯ iT 1C¯ i + A P¯¯ i P¯¯ i + εiC 0 0 εi
K¯ iT 1B¯ Ti 1B¯ i K¯ i 0
0 KiT 1BTi 1Bi Ki
¯
¯
+
−B¯ i Ki Ai − Li Ci
1 0 + C 0 εi
0 0
0 MLi
0 0
0
LTi M
T 2 ¯ ¯ P¯ P¯ + Q¯¯ + E¯¯ i Wi E¯¯ i B i i
εi ¯Ai + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki T P¯i 0 ≥ ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki T A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki P¯ i 0 + + Q¯¯ + E¯¯ i Wi E¯¯ i . ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki
(37)
T
Left and right multiplying the last argument in (37) by x¯¯ and x¯¯ , respectively, while considering (34) and (35), results in:
T x¯¯
T
A¯ i + 1A¯ i + B¯ i + 1B¯ i K¯ i − B¯ i + 1B¯ i Ki P¯ i 0 ˜ i − Li 1C¯ i + 1Bi K¯ i ∗ M 1 A A − L C − 1 B K i i i i i x¯¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ − Bi + 1Bi Ki Ai + 1Ai + Bi + 1Bi Ki P¯ 0 + Q¯¯ i + i ∗ M 1A˜ i − Li 1C¯ i + 1Bi K¯ i Ai − Li Ci − 1Bi Ki
T T < − E¯¯ i x¯¯ Wi E¯¯ i x¯¯ = − E¯ i x¯ Wi E¯ i x¯
(38)
T
given the fact that Wi has non-negative entries and considering (1), it is deduced that E¯ i x¯ Wi E¯ i x¯ > 0 for x ∈ Ωi , i ∈ I , thus the left hand side of (38) remains negative definite for x ∈ Ωi , i ∈ I . Now considering (31) and implementing the Schur Complement on (30c) results in:
RTi LTi MLi Ri < Xi .
(39)
The inequalities in (38) and (39), alongside (30d), imply that the inequality constraint in (33) is satisfied. Thus the proof is completed. Remark 5. The parameters and matrices to be determined by solving the constrained optimization problem in Theorem 1 include P¯ i , K¯ i , Yi , Xi , Ui ≻ 0, Wi ≻ 0, εiA > 0, εiB > 0, and εiC > 0, for i ∈ I and d and M. It should be noted that, in addition to the constraints mentioned throughout Theorem 1, the positivity of εiA , εiB and εiC , for i ∈ I, and the element-wise positivity of Ui and Wi , for i ∈ I, must be enforced while solving the constrained optimization problem in Theorem 1. Remark 6. It is notable that Theorem 1 only provides sufficient condition for the second moment stabilizability of the closed loop dynamics (15). In other words if the proposed design method in Theorem 1 fails to produce feasible observer–controller gain values one may not comment on the non-observability or non-controllability of the closed loop dynamics of (15). Remark 7. The observer–controller in (12) and (13) is capable of second moment stabilizing the uncertain stochastic piecewise affine system in (9) only if the information regarding the discrete state of the system (i.e. which one of the subspaces is occupied by the system at each given moment in time) are available and the observer–controller gain values may switch accordingly. Since in most practical implementations access to the exact discrete state of the system is not possible due to the presence of measurement noise, one must resort to a discrete state observer to work in conjunction with the proposed hybrid observer–controller in order to stabilize the system, see for example [30,31]. However, providing an exact mathematical proof for the stability of the continuous time observer error dynamics with the discrete state observer is very intricate and to the best of the authors’ knowledge remains as an open problem to this date. It is notable that the emphasis of this paper is on obtaining the stabilizing observer and controller gain values while designing a discrete state observer that guarantees the stability of the system with the PWA observer–controller is left for future researches.
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H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167 Table 1 Parameters of the helicopter model from [32]. Parameter
Value
Unit
Iyy mheli lcgx lcgz Fv M Fv M
0.0283 0.9941 0.0134 0.0289 0.0041 9.81
kg m2 kg m m N m/rad/s m/s2
4. Simulation results In this section, the observer–controller design algorithm proposed in the previous section is implemented on a simplified helicopter pitch dynamics model presented in [32]. The constrained optimization problem arising in Theorem 1 is solved using the solver PENBMI [33] interfaced via the toolbox YALMIP [34]. The state space model of the pitch and pitch rate dynamic equations of the helicopter model are given below:
d
x1 x2
=1
Iyy
x2 −mheli lcgx g cos (x1 ) dt + Sdw −mheli lcgz g sin (x1 ) −FvM x2 + u
(40a)
dz =
x1 dt + Rdv x2
(40b)
where x1 and x2 represent the pitch angle and pitch rate respectively. Iyy is the second moment around y-axis, mheli is the mass of helicopter, lcgx and lcgz are distance from center of mass relative to Joint B (see Fig. 1), Fv M is the pitch damping, and u is the control torque exerted by the main blade of helicopter around y-axis. The values of the parameters are shown in Table 1. It is evident from (40) that the system nonlinearity is only dependent on x1 and may be formulated as: f¯ (x1 ) =
−mheli g Iyy
lcgx cos (x1 ) + lcgz sin (x1 ) .
(41)
In order to find a piecewise linear approximation of the nonlinear dynamics, the state space of the nonlinear system is partitioned into the following subspaces:
9π 2π 2π Ωi = ( x 1 , x 2 ) | − + i ≤ x1 ≤ −π + i , 7
7
7
i = 1, . . . , 7.
(42)
Using a uniform grid of points, the piecewise linear approximation of the system nonlinearity, f¯ ∗ (x1 ), in the subspaces not containing the origin, is calculated such that the sum of the squared error between f¯ (x1 ) and f¯ ∗ (x1 ) is minimized and the approximation remains C 0 . In the subspace containing the origin, the linearization of the nonlinear dynamics at the origin is utilized as f¯ ∗ (x1 ). As mentioned before the uncertainty in the approximated PWA system arises in the form of the difference between f¯ (x1 ) and f¯ ∗ (x1 ). Denoting the squared error between f¯ (x1 ) and f¯ ∗ (x1 ) by g¯ (x1 ) and considering (11a) one may conclude: g¯ (x1 ) = f¯ (x1 ) − f¯ ∗ (x1 )
=
N
2
βi (x) (f (x) − Ai x − ai )T (f (x) − Ai x − ai )
i =1
=
N
βi (x) (1Ai x + 1ai )T (1Ai x + 1ai )
i =1
≤
N
βi (x) x¯ T Ei2A¯ x¯ = g¯ ∗ (x1 )
(43)
i =1
where g¯ ∗ (x1 ) is the piecewise quadratic upper bound of g¯ (x1 ) and βi (x) is the same as defined in (18). Given the positivity of g¯ (x1 ) , g¯ ∗ (x1 ) is easily obtained. The system nonlinearity and its piecewise linear approximation are demonstrated in Fig. 2 alongside g¯ (x1 ) and g¯ ∗ (x1 ). The augmented system matrices of the approximated PWA system are given below:
0 ¯A1 = 6.968 0
1 −0.145 0
0 27.028 , 0
1.712
E12A¯
= ∗
0 0
4.753 0 13.240
(44a)
H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
163
Fig. 1. Pitch model of the helicopter from [32].
Fig. 2. (a) System nonlinearity of simplified helicopter dynamics and its PWA approximation, (b) Squared error between system nonlinearity (g¯ (x)) and its quadratic upper bound (g¯ ∗ (x1 )).
A¯ 2 =
0 −2.128 0
0 ¯A3 = −10.728 0
1 −0.145 0 1 −0.145 0
0 6.617 , 0
E22A¯
0 −4.963 , 0
2.438
= ∗
1.760
E32A¯
0 0
= ∗
4.439 0 8.174
0 0
1.397 0 1.213
(44b)
(44c)
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H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
0 −9.959 0
1 −0.145 0
0 −4.618 , 0
E42A¯
0 ¯A5 = −2.846
1 −0.145 0
0 −7.810 , 0
E52A¯
A¯ 4 =
0
0 ¯A6 = 7.172
1 −0.145 0
0
0 ¯A7 = 11.152
35.336
B¯ i = 0 C¯ i =
1 0
0 1
T
0
0 , 0
0 −30.230 , 0
=
EiB¯ = 0,
0 0 0
0 0
−1.904
∗ 1.552
=
0 0
∗
Ei2C¯ = 0,
2.402
E62A¯
0 0
∗
,
1.822
=
0 −21.298 , 0
1 −0.145 0
0
E72A¯
0.290
= ∗
0 0
(44d)
0 1.592
−3.192
(44e)
0 6.603
(44f)
−0.811 0 2.267
(44g) (44h)
i∈I
i ∈ I.
(44i)
The identity matrix is selected for the weighting matrices, Qx and Qe . For the process noise matrix, Si , and sensor noise matrix, Ri , it is assumed that, 0.6 Si = 0
0 , 0.6
0 Ri = , 0.1
i ∈ I.
(45)
Since there are no uncertainties in the augmented input and output matrices, (30b) may be reduced to:
(1, 1) ∗
P¯¯ i
−εiA ¯I
< 0.
As mentioned before, the local closed loop equilibrium point of each subspace must be selected a priori. Apart from the subspace containing the origin, the equilibrium points for the rest of the subspaces are chosen outside their respective regions so that when the state of the system enters a region not containing the origin, it will eventually leave that region and is stabilized in the region containing the origin. eq
eq
eq
eq
x1 = −x7 = x3 = −x5 =
4.4880 , 0
x2 = −x6 =
0.8976 , 0
x4 =
eq
eq
eq
2.6928 , 0
(46)
0 . 0
Using the system matrices presented in (44) and (45), and the local closed loop equilibrium points assigned in (46), Theorem 1 is utilized to calculate the observer and controller gain values of (12) and (13) and the matrices allocated with the Lyapunov function given in (17). The maximum bound of the second moment stability converges to a value of d = 1.8143 which implies that E xT x + eT e < 1.8143 as time approaches infinity. The rest of the results are presented below:
−2.4690 T ¯K1 = −2.1704 , 9.4307 −3.2076 T K¯ 2 = −1.8225 , 8.6123 −7.5494 T K¯ 3 = −2.3830 , 7.1893 −9.9999 T ¯K4 = −2.5338 , 0.1307 −3.2276 T ¯K5 = −2.2282 , −2.7484
80.2375 L1 = −37.6647
13.6834 4.8802
(47a)
99.4575 −51.4419
3.1721 11.2601
(47b)
169.3088 −92.8010
−5.6440 14.1838
(47c)
6.2453 2.9793
(47d)
L2 =
L3 =
204.1161 L4 = −107.5031
303.6090 316.7910
L5 =
0.3293 0.6848
(47e)
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165
Fig. 3. Lyapunov function, (a) Piecewise quadratic segment allocated with system states, (b) Quadratic segment allocated with observer errors.
K¯ 6
K¯ 7
−2.9168 T = −2.6143 , −6.7050 −1.9400 T = −2.5707 , −6.4349
1.0047
P¯ 1 =
79.1474 L6 = −37.2775
13.2070 4.9029
(47f)
77.5300 L7 = −34.3340
19.5708 . 1.6702
(47g)
ε1A = 6.5982
(48a)
ε2A = 4.5979
(48b)
ε3A = 3.9556
(48c)
0.0053 0.0105
∗ 1.3747
P¯ 2 =
0.0072 0.0105
∗ 2.8319
P¯ 3 =
0.0129 0.0105
∗ 4.2439
P¯ 4 =
0.0388 0.0105
∗ 1.6984
P¯ 5 =
1.0156
P¯ 6 =
0.7262
P¯ 7 =
0.0919
∗
(48d)
ε5A = 2.9714
(48e)
0.0072 0.0105
2.7348 0.0194 , −2.8773
ε6A = 3.9209
(48f)
0.0053 0.0105
3.2592 0.0237 , −3.7731
ε7A = 6.4446
(48g)
∗ M =
ε4A = 2.1379
1.5245 0.0116 , −0.8556
∗
0 0 , 0
0.0129 0.0105
∗
−4.5091 −0.0237 , −4.2395 −3.7017 −0.0194 , −2.4784 −2.5419 −0.0116 , −1.9972
0.0995 . 0.1718
(48h)
The level surfaces of the obtained Lyapunov function are presented in Fig. 3 which exhibits its positive definiteness. Thus Theorem 1 shows that the closed loop system with the designed PWA observer–controller is stochastically stable with given upper bound on the second moment.
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H. Razavi et al. / Nonlinear Analysis: Hybrid Systems 19 (2016) 153–167
Fig. 4. Simulated Trajectories, (a) PWA system with PWA Obs/PWA Ctrl traj. 1, (b) PWA system with PWA Obs/PWA Ctrl traj. 2, (c) Nonlinear system with PWA Obs/PWA Ctrl traj. 1, (d) Nonlinear system with PWA Obs/PWA Ctrl traj. 2, (e) Nonlinear system with EKBF/PWA Ctrl traj. 1, (f) Nonlinear system with EKBF/PWA Ctrl traj. 2, (g) Nonlinear system with UKBF/PWA Ctrl traj. 1, (h) Nonlinear system with UKBF/PWA Ctrl traj. 2.
Table 2 Simulation results.
System
Controller
Observer
E xT x
PWA Nonlinear Nonlinear Nonlinear
PWA PWA PWA PWA
PWA PWA EKBF UKBF
0.7202 1.0757 2.1956 1.6495
E eT e
0.0750 0.1087 0.2650 0.1959
E xT x + eT e
0.7952 1.1844 2.4606 1.8454
The designed PWA observer–controllers are then applied on both the piecewise approximation model and the original nonlinear system for the following different initial conditions:
−3.00 , 1.50 3.00 Initial Condition 2: x0 = , −1.50
Initial Condition 1: x0 =
−2.85 , 1.25 2.85 xˆ 0 = . −1.25
xˆ 0 =
(49)
For comparison purposes, an extended Kalman–Bucy filter (EKBF) and an unscented Kalman–Bucy filter (UKBF), both in combination with the designed PWA controller, are applied on the nonlinear system. The initial conditions for the combined EKBF/PWA controller and the combined UKBF/PWA controller are the same as in (49) and the initial conditional covariance matrix for the EKBF and UKBF is assumed to be P0 = 0.05I. Utilizing the notation introduced in [35] the tuning parameters for the UKBF are set as α = 1, β = 2 and κ = 1. The system trajectories in phase space for 50 s are given in Fig. 4. As can be seen in Fig. 4, the piecewise approximation model and the nonlinear system responses are very similar which indicates that the approximation is close to the nonlinear system and the designed observer–controllers have very similar effects on both systems. As explained in Remark 1, the system is ergodic, thus the time average of xT x + eT e may be utilized to verify
the calculated value of d. The calculated time average of xT x + eT e using both the piecewise affine and nonlinear system trajectories are given in Table 2, both of which comply to the maximum calculated value of d = 1.8143. Other numerical results presented in Table 2 show that the PWA observer–controller has performed more efficiently compared to both the combined EKBF/PWA controller and the combined UKBF/PWA controller.
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167
5. Conclusion The problem of minimizing the bound on the variance of the steady response of a class of uncertain piecewise affine systems, subject to system and measurement additive noises, by means of a piecewise affine observer–controller, is formulated as an optimization problem subject to a number of constraints in the form of matrix inequalities. As a practical example, the proposed control scheme is utilized to stabilize the pitch dynamics of a simplified helicopter model. The nonlinear system dynamics are first approximated with an uncertain PWA system where the uncertainties arise in the form of the difference between the actual nonlinear dynamics and the PWA approximation. Simulation results show the efficiency of the control scheme in dealing with the effects of system uncertainties while minimizing the effects of external disturbances on the steady response. The performance of the PWA observer–controller is compared with an extended Kalman–Bucy filter and unscented Kalman–Bucy filter both in combination with the PWA controller. In addition to the fact that the stability of the system is only guaranteed in the case of the PWA observer–controller, it is observed that the PWA observer–controller has performed more efficiently compared to both the combined EKBF/PWA controller and the combined UKBF/PWA controller. Future research includes designing a hybrid discrete state observer to work in conjunction with the proposed PWA observer–controller. References [1] D. Mignone, G. Ferrari-Trecate, M. Morari, Stability and stabilization of piecewise affine and hybrid systems: An LMI approach, in: Proceedings of the 39th IEEE Decision and Control, Vol. 1, IEEE, 2000, pp. 504–509. [2] L. Rodrigues, A. Hassibi, J. How, Output feedback controller synthesis for piecewise-affine systems with multiple equilibria, in: Proceedings of American Control Conference, Vol. 3, IEEE, 2000, pp. 1784–1789. [3] L. Rodrigues, S. Boyd, Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization, Systems Control Lett. 54 (9) (2005) 835–853. [4] A. Hassibi, S. Boyd, Quadratic Stabilization and Control of Piecewise-Linear Systems, Vol. 6, IEEE, 1998, pp. 3659–3664. [5] A. Rantzer, M. Johansson, Piecewise linear quadratic optimal control, IEEE Trans. Automat. Control 45 (4) (2000) 629–637. [6] L. Rodrigues, J. How, Observer-based control of piecewise-affine systems, Internat. J. Control 76 (5) (2003) 459–477. [7] A. Ben Abdallah, M. Hammami, J. Kallel, Robust stability of uncertain piecewise-linear systems: LMI approach, Nonlinear Dynam. 63 (1–2) (2011) 183–192. [8] R. Oliveira, P. Peres, LMI conditions for the existence of polynomially parameter-dependent Lyapunov functions assuring robust stability, in: 44th IEEE Conference on Decision and Control, pp. 1660–1665. [9] J. Xu, L. Xie, Homogeneous polynomial Lyapunov functions for piecewise affine systems, in: Proceedings of the 2005 American Control Conference, pp. 581–586. [10] G. Feng, Controller design and analysis of uncertain piecewise-linear systems, IEEE Trans. Circuits Syst. I 49 (2) (2002) 224–232. [11] H. Zhang, G. Feng, C. Dang, Stability analysis and H∞ control for uncertain stochastic piecewise-linear systems, IET Control Theory Appl. 3 (8) (2009) 1059–1069. [12] M. Bujorianu, J. Lygeros, Toward a general theory of stochastic hybrid systems, Stochastic Hybrid Syst. (2006) 3–30. [13] D. Dimarogonas, K. Kyriakopoulos, Lyapunov-like stability of switched stochastic systems, in: Proceedings of American Control Conference, Vol. 2, IEEE, 2004, pp. 1868–1872. [14] A. Abate, L. Shi, S. Simic, S. Sastry, A stability criterion for stochastic hybrid systems, in: Mathematical theory of Networks and Systems Conference, 2004. [15] J. Xiong, J. Lam, Z. Shu, X. Mao, Stability analysis of continuous-time switched systems with a random switching signal. [16] C. Yuan, J. Lygeros, On the exponential stability of switching diffusion processes, IEEE Trans. Automat. Control 50 (9) (2005) 1422–1426. [17] Y. Dong, J. Sun, Brief paper: On hybrid control of a class of stochastic non-linear Markovian switching systems, Automatica 44 (4) (2008) 990–995. [18] E. Boukas, Static output feedback control for stochastic hybrid systems: LMI approach, Automatica 42 (1) (2006) 183–188. [19] D. De Farias, J. Geromel, J. Do Val, O. Costa, Output feedback control of Markov jump linear systems in continuous-time, IEEE Trans. Automat. Control 45 (5) (2000) 944–949. [20] L. Rodrigues, N. Gollu, Analysis and state feedback control for PWA systems with additive noise, in: Proceedings of American Control Conference, IEEE, 2006, pp. 5438–5443. [21] Z. Wu, M. Cui, P. Shi, H.R. Karimi, Stability of stochastic nonlinear systems with state-dependent switching, IEEE Trans. Automat. Control 58 (8) (2013) 1904–1918. [22] S. Battilotti, Observer design for nonlinear systems with Markov chain, Internat. J. Robust Nonlinear Control 19 (14) (2009) 1603–1631. [23] J.H. Ahrens, H.K. Khalil, High-gain observers in the presence of measurement noise: A switched-gain approach, Automatica 45 (4) (2009) 936–943. [24] J. Le Ny, E. Feron, M. Dahleh, Scheduling Kalman Filters in Continuous Time, IEEE, 2009, pp. 3799–3805. [25] K. Merat, H. Salarieh, A. Alasty, A. Meghdari, Stochastic piecewise affine control with application to pitch control of helicopter, Nonlinear Anal. Hybrid Syst. 15 (2015) 86–97. [26] S. Aberkane, J. Ponsart, M. Rodrigues, D. Sauter, Output feedback control of a class of stochastic hybrid systems, Automatica 44 (5) (2008) 1325–1332. [27] F. Xi, G. Yin, The strong feller property of switching jump-diffusion processes, Statist. Probab. Lett. 83 (3) (2013) 761–767. [28] M. Zakai, A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise, SIAM J. Control 7 (1969) 390. [29] I.R. Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems Control Lett. 8 (4) (1987) 351–357. [30] C.E. Seah, I. Hwang, State estimation for stochastic linear hybrid systems with continuous-state-dependent transitions: an IMM approach, IEEE Trans. Aerosp. Electron. Syst. 45 (1) (2009) 376–392. [31] C.E. Seah, I. Hwang, Algorithm for performance analysis of the IMM algorithm, IEEE Trans. Aerosp. Electron. Syst. 47 (2) (2011) 1114–1124. [32] R. Endreß, L. Rodrigues, Modeling and control of two degrees of freedom quanser helicopter, Tech. Rep., Concordia University, 2008. [33] M. Kočvara, M. Stingl, Pennon: A code for convex nonlinear and semidefinite programming, Optim. Methods Softw. 18 (3) (2003) 317–333. [34] J. Lofberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in: IEEE International Symposium on Computer Aided Control Systems Design, IEEE, 2004, pp. 284–289. [35] S. Sarkka, On unscented Kalman filtering for state estimation of continuous-time nonlinear systems, IEEE Trans. Automat. Control 52 (9) (2007) 1631–1641.