Discrete Time Systems Part 8 pot

Then for all admissible uncertainties and possible faults M , the faulty closed-loop system 6 with satisfactory fault-tolerant H∞ performance and cost function indices for fault-toleran

Quadratic D Stabilizable Satisfactory Fault-tolerant Control with

Constraints of Consistent Indices for Satellite Attitude Control Systems 199

Theorem 2: Consider the system (1) and the cost function (7), for the given index ( , )Φq r

and H∞ norm-bound index γ , if there exists symmetric positive matrix X , matrix Y and

scalars εi>0(i=4 ~ 9) such that the following linear matrix inequality

holds, where Σ21=[(CX EY+ )T, , ,X X Y Y Y Y Y Y J , T, T, T, T, T, T ] Σ22=diag[− +I ε7EJET,−Q−1,−ε4I ,

− I+ J− J − J − J I R− − I Then for all admissible uncertainties and

possible faults M , the faulty closed-loop system (6) with satisfactory fault-tolerant

H∞ performance and cost function indices for fault-tolerant control is deduced as the

following optimization problem

Theorem 3: Given quadratic D stabilizability index ( , )Φ q r , suppose the system (1) is robust

fault-tolerant state feedback assignable for actuator faults case, then LMIs (10), (13) have a

feasible solution Thus, the following minimization problem is meaningful

min γ : X Y, , ,γ εi S.t LMIs (10), (13) (14)

Proof: Based on Theorem 1, if the system (1) is robust fault-tolerant state feedback

assignable for actuator faults case, then inequality

0

T

C− <

A PA P has a feasible solution P , K And existing λ> , 00 δ> , the following inequality holds

0

λ⎡⎣A PA −P⎤⎦+C C +Q K MRMK+ +δI< (15) Then existing a scalar γ0, when γ γ> 0, it can be obtained that

Using Schur complement and Theorem 2, it is easy to show that the above inequality is

equivalent to linear matrix inequality (13), namely, P1, K , γ is a feasible solution of LMIs

Discrete Time Systems

200

(10), (13) So if the system (1) is robust fault-tolerant state feedback assignable for actuator

faults case, the LMIs (10), (13) have a feasible solution and the minimization problem (14) is

meaningful The proof is completed

Suppose the above minimization problem has a solution XL, YL, εiL, γL, and then any

index γ γ> L, LMIs (10), (13) have a feasible solution Thus, the following optimization

problem is meaningful

Theorem 4: Consider the system (1) and the cost function (7), for the given quadratic D

stabilizability index ( , )Φq r and H∞ norm-bound index γ γ> L, if there exists symmetric

positive matrix X , matrix Y and scalars εi>0(i=1 ~ 9) such that the following

minimization

2 2

S.t (i) (10), (13) (ii) ⎡−λ T⎤ 0

u Kx M Y X x is an optimal guaranteed cost satisfactory fault-tolerant

controller, so that the faulty closed-loop system (6) is quadratically D stabilizable with an H∞

norm-bound γ , and the corresponding closed-loop cost function (7) satisfies

2 2

min

J≤λ +γ β

According to Theorem 1~4, the following satisfactory fault-tolerant controller design

method is concluded for the actuator faults case

Theorem 5: Given consistent quadratic D stabilizability index ( , )Φ q r , H∞ norm index

u Kx M YX x is satisfactory fault-tolerant controller making the faulty

closed-loop system (6) satisfying the constraints (a), (b) and (c) simultaneously

In a similar manner to the Theorem 5, as for the system (1) with quadratic D stabilizability,

H∞ norm and cost function requirements in normal case, i.e., M I , we can get the =

satisfactory normal controller without fault tolerance

Suppose the actuator failure parameters Ml=diag{0.4, 0.6}, Mu=diag{1.3, 1.1} Given the

quadratic D stabilizability index (0.5,0.5)Φ , we can obtain state-feedback satisfactory

fault-tolerant controller (SFTC), such that the closed-loop systems will meet given indices

constraints simultaneously based on Theorem 5

Quadratic D Stabilizable Satisfactory Fault-tolerant Control with

Constraints of Consistent Indices for Satellite Attitude Control Systems 201

SFTC

0.5935 3.01876.7827 5.6741

of closed-loop system driven by normal controller lie in the circular disk Φ(0.5,0.5) for normal case (see Fig 1) However, in the actuator failure case, the closed-loop system with normal controller is unstable; some poles are out of the given circular disk (see Fig 2) In the contrast, the performance by satisfactory fault-tolerant controller still satisfies the given pole index (see Fig 3) Thus the poles of closed-loop systems lie in the given circular disk by the proposed method

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Fig 1 Pole-distribution under satisfactory normal control without faults

5 Conclusion

Taking the guaranteed cost control in practical systems into account, the problem of satisfactory fault-tolerant controller design with quadratic D stabilizability and H∞ norm-bound constraints is concerned by LMI approach for a class of satellite attitude systems subject to actuator failures Attention has been paid to the design of state-feedback controller that guarantees, for all admissible value-bounded uncertainties existing in both the state and control input matrices as well as possible actuator failures, the closed-loop system to satisfy

Discrete Time Systems

202

the pre-specified quadratic D stabilizability index, meanwhile the H∞ index and cost function are restricted within the chosen upper bounds So, the resulting closed-loop system can provide satisfactory stability, transient property, H∞ performance and quadratic cost performance despite of possible actuator faults The similar design method can be extended

to sensor failures case

Fig 3 Pole-distribution under satisfactory fault-tolerant control with faults

Quadratic D Stabilizable Satisfactory Fault-tolerant Control with

Constraints of Consistent Indices for Satellite Attitude Control Systems 203

6 Acknowledgement

This work is supported by the National Natural Science Foundation of P R China under grants 60574082, 60804027 and the NUST Research Funding under Grant 2010ZYTS012

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D Zhang, Z Wang, and S Hu Robust satisfactory fault-tolerant control of uncertain linear

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Part 3

Discrete-Time Adaptive Control

Chenguang Yang1and Hongbin Ma2

On the other hand, the early studies on adaptive control were mainly concerning on theparametric uncertainties, i.e., unknown system parameters, such that the designed controllaws have limited robustness properties, where minute disturbances and the presence ofnonparametric model uncertainties can lead to poor performance and even instability ofthe closed-loop systems Egardt (1979); Tao (2003) Subsequently, robustness in adaptive

the difficulties associated with discrete-time uncertain nonlinear system model, there areonly limited researches on robust adaptive control to deal with nonparametric nonlinearmodel uncertainties in discrete-time systems For example, in Zhang et al (2001), parameterprojection method was adopted to guarantee boundedness of parameter estimates in presence

of small nonparametric uncertainties under certain wild conditions For another example,the sliding mode method has been incorporated into discrete-time adaptive control Chen(2006) However, in contrast to continuous-time systems for which a sliding mode controllercan be constructed to eliminate the effects of the general uncertain model nonlinearity, fordiscrete-time systems, the uncertain nonlinearity is normally required to be of small growthrate or globally bounded, but sliding mode control is yet not able to completely compensatefor the effects of nonlinear uncertainties in discrete-time As a matter of fact, unlike incontinuous-time systems, it is much more difficulty in discrete-time systems to deal with

Discrete-Time Adaptive Predictive Control

with Asymptotic Output Tracking

13

nonlinear uncertainties When the size of the uncertain nonlinearity is larger than a certainlevel, even a simple first-order discrete-time system cannot be globally stabilized Xie & Guo(2000) In an early work on discrete-time adaptive systems, Lee (1996) it is also pointedout that when there is large parameter time-variation, it may be impossible to construct

a global stable control even for a first order system Moreover, for discrete-time systems,most existing robust approaches only guarantee the closed-loop stability in the presence ofthe nonparametric model uncertainties, but are not able to improve control performance bycomplete compensation for the effect of uncertainties

Towards the goal of complete compensation for the effect of nonlinear model uncertainties

in discrete-time adaptive control, the methods using output information in previous steps

to compensate for uncertainty at current step have been investigated in Ma et al (2007)for first order system, and in Ge et al (2009) for high order strict-feedback systems Wewill carry forward to study adaptive control with nonparametric uncertainty compensation

for NARMA system (nonlinear auto-regressive moving average), which comprises a general

nonlinear discrete-time model structure and is one of the most frequently employed form indiscrete-time modeling process

Time delay is an active topic of research because it is frequently encountered in engineeringsystems to be controlled Kolmanovskii & Myshkis (1992) Of great concern is the effect

with time delays, some of the useful tools in robust stability analysis have been welldeveloped based on the Lyapunov’s second method, the Lyapunov-Krasovskii theorem andthe Lyapunov-Razumikhin theorem Following its success in stability analysis, the utility

of Lyapunov-Krasovskii functionals were subsequently explored in adaptive control designsfor continuous-time time delayed systems Ge et al (2003; 2004); Ge & Tee (2007); Wu (2000);Xia et al (2009); Zhang & Ge (2007) However, in the discrete-time case there dos not exist

a counterpart of Lyapunov-Krasovskii functional To resolve the difficulties associated withunknown time delayed states and the nonparametric nonlinear uncertainties, an augmented

states vector is introduced in this work such that the effect of time delays can be canceled atthe same time when the effects of nonlinear uncertainties are compensated.

In the NARMA system described in (1), we can see that there is a “relative degree” n which can be regarded as response delay from input to output Thus, the control input at the kth

and ideally the controller should also incorporate the information of these states However,dependence on these future states will make the controller non-causal!

Diophantine function by using which system (1) can be transformed into an n-step predictor

control can be designed under certainty equivalence principal to emulate a deadbeat controller, which forces the n-step ahead future output to acquire a desired reference value However, transformation of the nonlinear system (1) into an n-step predictor form would make the

known nonlinear functions and unknown parameters entangled together and thus not

identifiable Thus, we propose future outputs prediction, based on which adaptive control can

be designed properly

Throughout this chapter, the following notations are used

3 Assumptions and preliminaries

Some reasonable assumptions are made in this section on the system (1) to be studied Inaddition, some useful lemmas are introduced in this section to facilitate the later controldesign

Assumption 3.1. In system (1), the functional uncertainty ν(·), satisfies Lipschitz condition, i.e.,

ν(ε1) −ν(ε2) ≤ L νε1−ε2, ∀ε1,ε2 ∈ R n , where L ν < λ∗ with λ∗ being a small number defined in (58) The system functions φ i(·), i=1, 2, , n, are also Lipschitz functions with Lipschitz

209

Discrete-Time Adaptive Predictive Control with Asymptotic Output Tracking

models are derived from continuous-time models, the growth rate of nonlinear uncertaintycan always be made sufficient small by choosing sufficient small sampling time.

Assumption 3.2. In system (1), the control gain coefficient g m of current instant control input u(k)

is bounded away from zero, i.e., there is a known constant g m>0 such that|g m| >g m , and its sign is known a priori Thus, without loss of generality, we assume g m>0.

Remark 3.2. It is called unknown control direction problem when the sign of the control gain is unknown The unknown control direction problem of nonlinear discrete-time system has been well addressed in Ge et al (2008); Yang et al (2009) but it is out the scope of this chapter.

Definition 3.1. Chen & Narendra (2001) Let x1(k)and x2(k)be two discrete-time scalar or vector signals,∀k∈Z+t , for any t.

• We denote x1(k) =O[x2(k)], if there exist positive constants m1, m2and k0such thatx1(k) ≤

m1maxk≤kx2(k) +m2, ∀k>k0

• We denote x1(k) =o[x2(k)], if there exists a discrete-time function α(k)satisfying lim k→∞α(k) =

• We denote x1(k) ∼x2(k)if they satisfy x1(k) =O[x2(k)]and x2(k) =O[x1(k)].

Assumption 3.3. The input and output of system (1) satisfy

u(k) =O[y(k+n)] (5)Assumption 3.3 implies that the system (1) is bounded-output-bounded-input (BOBI) system(or equivalently minimum phase for linear systems)

According to Definition 3.1, we have the following proposition

Proposition 3.1. According to the definition on signal orders in Definition 3.1, we have following properties:

(vi) If x1(k) ∼x2(k)and lim k→∞x2(k) =0, then lim k→∞x1(k) =0.

(vii) If x1(k) =o[x1(k)] +o[1], then lim k→∞x1(k) =0.

(viii) Let x2(k) =x1(k) +o[x1(k)] If x2(k) =o[1], then lim k→∞x1(k) =0.

Proof.See Appendix A

Lemma 3.1. Goodwin et al (1980) (Key Technical Lemma) For some given real scalar sequences s(k),

b1(k), b2(k)and vector sequence σ(k), if the following conditions hold:

Then, ifZ(k)is bounded we haveΔZ(k) →0 as well asν(z(k−τ)) −ν(z(l k−τ)) →0.

Proof.Given the definition of l kin (7), it has been proved in Ma (2006); Xie & Guo (2000) that

4 Future output prediction

In this section, an approach to predict the future outputs in (4) is developed to facilitate controldesign in next section To start with, let us define an auxiliary output as

y a(k+n−1) = ∑n

i=1θ T

i φ i(y(k+n−i)) +ν(z(k−τ)) (11)such that (1) can be rewritten as

For convenience, we introduce the following notations

Step 2: By using the estimates ˆθ i (k) and ˆg j(k)and according to (16), the two-step ahead future

which will be used for prediction in next step and where

Continuing the procedure above, we have three-step ahead future output prediction and so

Step (n−1): The(n−1)-step ahead future output is predicted as

Remark 4.1. Note that ˆ θ i(k−n+l+1)and ˆg j(k−n+l+1)instead of ˆ θ i(k)and g j(k)are used

in the prediction law of the l-step ahead future output In this way, the parameter estimates appearing

in the prediction of ˆy(k+l|k)and ˆy(k+l|k+1)are at the same time step, such that the analysis of prediction error will be much simplified.

Remark 4.2. Similar to the prediction procedure proposed in Yang et al (2009), the future output prediction is defined in such a way that the j-step prediction is based on the previous step predictions The prediction method Yang et al (2009) is further developed here for the compensation of the effect

of the nonlinear uncertainties ν(z(k−τ)) With the help of the introduction of previous instant l k defined in (7), it can been seen that in the transformed system (16) that the output information at previous instants is used to compensate for the effect of nonparametric uncertainties ν(z(k−τ))at the current instant according to (15).

The parameter estimates in output prediction are obtained from the following update laws