10

H∞ Output-Feedback Control of Discrete-Time

Systems

We consider the following system

xk+1 = fk(xk, vk, uk, ωk) (10.1)

together with a measurement process:

yk = C(xk) +D(xk)vk +G2(xk)w2k

where {xk}k≥0 is a solution to (10.1), with: an initial condition x0, an ex-ogenous disturbances {vk}k≥0, a control signal {uk}k≥0, and a white noisesequence ω = {ωk}k≥0 defined on a probability space (Ω,F, P ). We assumethat the assumptions of subsection 1.3.4 are all satisfied. We also consider thefollowing controlled output

z(x, u) = col{h(x), D12u} (10.2)

where h : Rn → Rr, and DT12D12 = R > 0.

In Chapter 9 (Th. 9.1) a BRL condition has been derived that guaranteesa prescribed bound on the l2−gain of the system 10.1. As an application ofthis BRL we consider now the time-invariant version of (10.1), that is:

xk+1 = f(xk, vk, uk, ωk) (10.3)

with the controlled output (10.2) and with time-invariant h. In this case oneeasily establishes, analogously to the time-varying case, a BRL which is atrivial corollary of Theorem 9.1. It yields a time-invariant algebraic HJI, thecounterpart of the well known linear stochastic case. This BRL is given in thefollowing Corollary.

Corollary 10.0.1 Consider the system (10.3) with the controlled output(10.2), and let uk = 0 ∀k ≥ 0.

(a) Let γ > 0, and let V be a positive real valued function: V : Rn → R+

satisfying the following Hamilton–Jacobi Algebraic Inequality (HJAI)

E. Gershon & U. Shaked: Advanced Topics in Control & Estimation, LNCIS 439, pp. 165–176.DOI: 10.1007/978-1-4471-5070-1_10 c© Springer-Verlag London 2013

166 10 H∞ Output-Feedback Control of Discrete-Time Systems

V (x) ≥ supv∈Rn

{||z(x)||2 − γ2||v||2 + Eωk{V [f(x, v, ωk)]}} (10.4)

for some control signal {uk}k≥0, and for all x ∈ Rn. Assume also: E{V (x0)} ≤γ2E{||x0||2} for all F0-measurable random variables x0 with E{||x0||2} < ∞.Then, the system of (10.3) has l2-gain≤ γ.

(b) Assume there is a control signal {uk}k≥0 which makes the system(10.3) l2-gain≤ γ for some γ > 0 and for all F0-measurable x0 withE{||x0||2} < ∞. Then, there is a positive function V which satisfies the HJI(10.4).

We remark that exponential mean square stability in the time invariant caseis easily established under a proper adjustment of the conditions obtained forexponential mean square stability for the time varying case.

10.1 Sufficient Conditions for l2-Gain≤ γ: A SpecialCase

In this section a special case of (10.1) is considered which characterizes a largeclass of systems (e.g. mechanical systems). For this system we establish a moretractable sufficient condition in terms of a HJI. We consider the followingsystem:

xk+1 = f(xk) + g1(xk)vk + g2(xk)vkω2k +G(xk)ω

1k, z = h(xk) (10.5)

where {ω2k}k is an Rl-valued white noise sequence with uncorrelated com-

ponents defined on the probability space (Ω,F , P ), and with the covarianceE{ω1

k(ω1k)

T } = R = diag{r1, r2, ..., rl}. Also {ω1k}k is an R1-valued white

noise process, with the covariance r = E{(ω2k)

2}. The exogenous disturbance{vk}k≥0 is assumed to be {Fk}k≥0-adapted, that is vk is Fk−1-measurablefor all k, where Fk is the minimal σ-algebra generated by {ωi : i ≤ k − 1}.Moreover, it is assumed that

∑∞k=1 ||vk||2 <∞. The functions fk : Rn → Rn,

g1k : Rn → Rn×m1 , g2k : Rn → Rn×m1 , and Gk : Rn → Rn×l are all assumedto be continuous. It is further assumed that the above functions are of thefollowing structure. Gk(x) is taken to be an n× l matrix, so that

Gk(x) = [G1k(x)x G2

k(x)x · · ·Glk(x)x] (10.6)

where Gik(x) are n× n matrices. It follows thatGk(x)ωk =

∑li=1G

ik(x)xω

ik.

It is also assumed that: fk(x) = Fk(x)x and hk(x) = Hk(x)x. Applicationof the BRL yields the following.

Lemma 10.1.1 Consider the system (10.5), with fk, g1k, g

2k, Gk, hk as defined

above(a) Let {Qk(x)}k≥0 be a family of symmetric positive matrices for all

x ∈ Rn satisfying

10.1 Sufficient Conditions for l2-Gain≤ γ: A Special Case 167

(i) φk(x)Δ= [γ2I − g1Tk (x)Qk+1(x)g

1k(x)− rg2Tk (x)Qk+1(x)g

2k(x)] > 0 (10.7)

for all x in Rn.(ii) Assume also that there is a positive function V : Rn → R+ with V (0) = 0which satisfies

Ewk{Vk[fk(x)+gk(x)u+g1k(x)v+G1

k(x)w1k]}−Vk(x)

≤Ewk{[fk(x)+gk(x)u + g1k(x)v +G1

k(x)w1k]

TQk(x)[fk(x) + gk(x)u + g1k(x)v

+G1k(x)w

1k ]} − xTQk(x)x

(10.8)and

xTQk(x)x > xT {FT

k (x)Qk+1(x)Fk(x) +l∑

i=1

ri(Gik(x))

TQk+1(x)Gik(x)

+FTk (x)Qk+1(x)g

1k(x)φ

−1k (x)g1Tk (x)Qk+1(x)Fk(x) +H

Tk (x)Hk(x)}x

(10.9)for all x in Rn, where ri = E(ω1

kiω1ki).

(iii) Q0(x) ≤ γ2I, ∀x ∈ Rn. (10.10)

Then, the system (10.5) has l2-gain≤ γ.(b) Assume the system (10.5) is dissipative with respect to the supply rate

γ2||v||2 − ||z||2 for some γ > 0, with a storage function Vi(x) = xTQi(x)x,where {Qi(x)}i≥0 are symmetric with Qi(x) ≥ 0 for all x ∈ Rn. Then, theinequality (10.9) holds.

Remark 10.1. Assumption (ii) in the above lemma is trivially satisfied in thecase where Qk is independent of the state vector. It can also be applied inthe case where the nonlinearities of (10.5) are of a polynomial type, for whichthe method of SOS (see, e.g [101]) can be used to obtain an equivalent problemwith a quadratic storage function (namely, with an augmented Qk that isindependent of the states).

In the case where g2k in (10.9) is identically zero, we apply the matrix inversionlemma to the third term in the right side of (10.9) and obtain the following.

Corollary 10.1.1 In the case where g2k is identically zero, the requirement of(10.9) becomes:

−Qk(x)+F Tk (x)[Q

−1k+1(x)−γ−2g1k(x)g

1Tk (x)]−1Fk(x)+

l∑i=1

ri(Gik(x))

TQk+1(x)Gik(x)

+HTk (x)Hk(x)<0

Consider now the time-invariant version of (10.5), that is:

xk+1 = f(xk) + g1(xk)vk + g

2(xk)vkω2k +G(xk)ω

1k, zk = h(xk) (10.11)

168 10 H∞ Output-Feedback Control of Discrete-Time Systems

where the assumptions associated with (10.5) now read

G(x) = [G1(x)x G2(x)x · · ·Gl(x)x] (10.12)

so that G(x)ωk =∑l

i=1Gi(x)xωi

k, f(x) = F (x)x and h(x) = H(x)x.

In this context, Eq. (10.7) and the inequality (10.9) of Lemma 10.1.1 reduceto

φ(x)Δ= [γ2I − g1T (x)Q(x)g1(x)− rg2T (x)Q(x)g2(x)] > 0 (10.13)

for all x in Rn. Assume also that there is a positive function V : Rn → R+

with V (0) = 0 which satisfies

Ewk{V [f(x)+g(x)u+g1(x)v+G1(x)w1

k]}−V (x)≤Ewk

{[f(x)+g(x)u+ g1(x)v +G1(x)w1k]

TQ(x)[f(x) + g(x)u+ g1(x)v

+G1(x)w1k]}−xTQ(x)x

(10.14)and

xTQ(x)x > xT {FT (x)Q(x)F (x) +l∑

i=1

ri(Gi(x))TQ(x)Gi(x)

+FT (x)Q(x)g1(x)φ−1(x)g1T (x)Q(x)F (x) +HT (x)H(x)}x(10.15)

for all x in Rn. In the case where g2 = 0 the latter inequality is satisfied if,

for all x in Rn,

−Q(x)+FT(x)[Q(x)−1−γ−2g1(x)g1T(x)]−1F (x)+l∑

i=1

ri(Gi(x))TQ(x)Gi(x)

+HT(x)H(x)<0(10.16)

The following example demonstrates the applicability of the algorithm of(10.13) and (10.15).

10.1.1 Example

We consider the following model for a single degree of freedom inverted pen-dulum with the multiplicative white noise ω:

mL2θ −mgL sin(θ) + (ς + ω)θ + κθ = u+ 2v (10.17)

where κ is the spring coefficient and ς is a damping coefficient. The signalv is a deterministic disturbance acting on the control input u and ω is astochastic uncertainty in the damping. This uncertainty stems from the facts

10.1 Sufficient Conditions for l2-Gain≤ γ: A Special Case 169

that damping is temperature dependent and that temperature possesses acomponent that can be modeled as a standard white-noise (r1 = 1) . In thismodel, θ is the inclination angle of the pendulum, L and m are its length andmass, respectively, and g is the gravitation coefficient.

Denoting x1 = θ and x2 = ml2θ, we apply Euler discretization to themodel of (10.17) and obtain the following discrete-time state-space represen-tation of the system.

⎡⎣x1x2

⎤⎦k+1

=

⎡⎣ 1 T

ml2

Tmglsin(x1,k)x1,k − κT 1− T ς

ml2

⎤⎦⎡⎣x1x2

⎤⎦k

+

⎡⎣ 0

T

⎤⎦uk

+

⎡⎣ 0

2T

⎤⎦ vk +

[0

−√Tx2,kml2

]ωk

where T is the sampling period. In this case l = 1 and we seek a constantstate-feedback control gain matrix K =

[k1 k2

]such that the control law

uk = Kxk achieves l2-gain ≤ γ for the closed-loop system that relates thedisturbance νk to the angle θk with a weight of 0.1 on the control effort uk,namely, zk = col{x1k, 0.1uk}.

Using the notation associated with (10.11) we have now f(x) = F (x)xwhere

F (x) = F (x) +

[0

T

]K, F (x)=

⎡⎣ 1 T

mL2

α(x)−κT 1− T ςmL2

⎤⎦, α(x)=TmgL sin(x1)x1 ,

g1(x) =

[02T

]; G1(x) =

[0 0

0 −√T

mL2

], and H =

[[1 0]0.1K

]

We require a minimum γ in the case where: m = 0.5kg, L = 0.7meter, κ =0.5Newton/meter and ς = 0.25 and where T = 0.01sec.

Considering a constant Q, the only function in (10.16) that depends onx is F (x). Applying Schur complement formula, Eqs. (10.16) and (10.7) arethen equivalent to the following matrix inequalities.

⎡⎢⎢⎢⎢⎣

−Q FT (x)Q 0 HT G1TQr1∗ −Q g1 0 0∗ ∗ −γ2I 0 0∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ −r1Q

⎤⎥⎥⎥⎥⎦ < 0,

[γ2I g1TQ∗ Q

]> 0. (10.18)

The latter two inequalities should be satisfied by Q and K for all x in Rn.The first one, (10.18a), is bilinear in Q and K. Note that, in fact, (10.18b)is included in (10.18a) and thus only the latter should be solved for. We,therefore, define P = Q−1 and Y = KP and multiply, from both sides,

170 10 H∞ Output-Feedback Control of Discrete-Time Systems

(10.18a), by diag{P, P, I, I, P}. We obtain the following LMI in the decisionvariables P and Y .

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P PFT (α) + Y T [0 T ] 0 P

[10

]0.1Y T PG1T r1

∗ −P g1 0 0 0∗ ∗ −γ2I 0 0 0∗ ∗ ∗ −I 0 0∗ ∗ ∗ 0 −I 0∗ ∗ ∗ ∗ ∗ −r1P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦< 0 (10.19)

where α is considered as an uncertain parameter that resides in the interval[0 TmgL]. Solving the latter LMIs at the two vertices (α = 0, TmgL) aminimum attenuation level γ = 0.2193 for K = −[22.5908 99.9143] (and

P =

[8.2656 −1.8945

∗ 8.7656

]).

10.2 The OF Case

In this section we develop an output-feedback controller. the development isbased on utilization of the bounded real lemma which has been establishedin the previous section. As only noisy partial state observations are available,we employ a nonlinear observer in the synthesis of the desired controller.

In what follows we consider a time invariant nonlinear system. This system,together with its observation process is given by

xk+1 = f(xk) + g1(xk)vk + g(xk)uk +G

1(xk)w1k

yk = C(xk) +D(xk)vk +G2(xk)w2k.

(10.20)

In addition, the controlled output z of (10.2) is considered. Here, y is Rr-valued, w1 and w2 are independent white noise stochastic processes which areassumed to be one dimensional each (this assumption is made for the sake ofconvenience only). The matrices C, D and G2 are taken to be of pertinentdimensions.

In order to establish the output-feedback case we need little more than theBRL as the central controller is to be utilized in the sequel rather than theone we have established in the preceding section. In fact we have the followingtheorem

Theorem 10.2.1 Consider the the system (10.20) together with the con-trolled output (10.2) and let x0 be an F0-measurable with E{||x0||2} < ∞.Assume there is a positive matrix Q(x) : Q(x) ≥ αI, ∀x ∈ Rn, for someα > 0, so that

E{xT0Qx0} ≤ γ2E{||x0||2}

10.2 The OF Case 171

for a prescribed positive scalar γ. Let φv(x)Δ= γ2I − g1T (x)Q(x)g1(x) and

assume that φv(x) > 0 for all x ∈ Rn. Define φu(x)Δ= R + gT (x)Q(x)g(x)

where Q(x) = Q(x)g1(x)φ−1v (x)g1T (x)Q(x) +Q(x), and let:

v∗(x) = φ−1v (x)g1T (x)Q(x)[f(x) + g(x)u∗(x)]

whereu∗(x) = −φ−1

u (x)gT (x)Q(x)f(x).Assume also that there is a positive function V : Rn → R+ with V (0) = 0

which satisfies

Ewk{V [f(x)+g(x)u+g1(x)v+G1(x)w1

k]}−V (x)≤Ewk

{[f(x)+g(x)u+ g1(x)v +G1(x)w1k]

TQ(x)[f(x) + g(x)u+ g1(x)v

+G1(x)w1k]} − xTQ(x)x

(10.21)

Then, a sufficient condition for the closed-loop system to have l2-gain≤ γ isthe existence of the following HJI:

hT (x)h(x)+fT (x)[Q(x)−Q(x)g(x)φ−1u (x)gT (x)Q(x)]f(x)−xTQx

+rxTG1T (x)Q(x)G1(x)x ≤ 0

for all x ∈ Rn.

Proof : Applying completion to squares, the following is readily obtained.

hT (x)h(x) + ||D12u||2 − γ2||v||2 − xTQ(x)x+Ewk

{[f(x) + g(x)u + g1(x)v +G1(x)w1k]

TQ(x)[f(x) + g(x)u+ g1(x)v

+G1(x)w1k]} = hT (x)h(x) + fT (x)[Q(x)− Q(x)g(x)φ−1

u (x)gT (x)Q(x)]f(x)

−xTQ(x)x + rxTG1(x)Q(x)G1(x) + [u−u∗]Tφu[u−u∗]− [v−v∗]Tφv[v−v∗](10.22)

where u∗ and v∗ are defined above. Letting now u = u∗ and v = v∗ and usingthe hypothesis of the theorem together with (10.21) yields

V (x)≥ infu∈Rn

supv∈Rm

{||z||2−γ2||v||2+Ewk{V [f(x)+g1(x)v+g(x)u+wk]}}, ∀x∈Rn

(10.23)which implies the conclusion of the theorem. ��For the special case where f(x) = F (x)x and h(x) = H(x)x for somefunctions F (x) and H(x) one has the following corollary which establishesa nonlinear matrix inequality as a sufficient condition for the closed-loop withstate-feedback to posses an l2-gain≤ γ.

Corollary 10.2.1 Suppose all the hypotheses of the last theorem hold, and letf(x) = F (x)x and h(x) = H(x)x. Then, the system (10.20) has l2-gain≤ γprovided the following inequality.

172 10 H∞ Output-Feedback Control of Discrete-Time Systems

HT (x)H(x) + FT (x)[Q(x)− Q(x)g(x)φ−1u (x)gT (x)Q(x)]F (x) −Q

+rG1T (x)Q(x)G1(x) ≤ 0(10.24)

holds for all x ∈ Rn.

Remark 10.2. The hypothesis regarding the structure of h(x) and f(x) as as-sumed in the last corollary is not a severe restriction as it is well known thatany differentiable function f(x) with f(0) = 0 can be written as f(x) = F (x)x

where F (x) =∫ 1

0 fx(tx)dt.

Utilizing the last theorem we proceed now in the direction of establishingan output-feedback controller which guarantees an l2-gain≤ γ of the closed-loop system (10.20). As a full state observation is assumed to be unavailablewe adopt a common practice for synthesizing the desired controller. In factwe assume a controller structure which is identical to the one established inTheorem 10.2.1 and which operates on the state-estimate that is given by

xk+1 = f(xk)+g1(xk)v

∗(xk)+g(xk)u∗(xk)+K(xk)[yk−C(xk)−D(xk)v∗(xk)](10.25)

where K(x) is the estimator gain, and has to be determined so that the closed-loop system achieves an H∞ performance. The controller is given by

u∗(x) = −φ−1u (x)gT (x)Q(x)f(x)

= −φ−1u (x)gT (x)Q(x)[I − γ−2g1T (x)g1(x)Q(x)]−1f(x),

and v∗(x) = φ−1v (x)g1T (x)Q(x)[f(x) + g(x)u∗(x)].

In order to facilitate the synthesis of the desired output-feedback closed-loop system, we consider the following augmented system.

xek+1 = fe(xek) + ge(xek)sk +G

e(xek)wek (10.26)

where

fe(xek)= col{f(xk)+g1(xk)v∗k(xk)+g(xk)u∗k(xk),

f(xk)+g(xk)u∗k(xk)+K(xk)[C(xk)−C(xk)] + K(x)[D(x)−D(x)]v∗k(xk)}

xek =

[xkxk

], ge(xek) =

[g1(xk)

K(xk)D(xk)

], Ge(xek) =

[G1(xk) 0

0 K(xk)G2(xk)

],

wek =

[w1

k

w2k

], sk = vk −v∗k(xk).

Having introduced the above augmented system, we are now ready to stateand prove the following theorem which establishes a solution to the output-feedback H∞ control of the underlying system.

Theorem 10.2.2 Consider the system (10.20) and (10.26). Let γ > 0 befixed and assume that there exist a positive function V (x) and a matrix Q :

10.2 The OF Case 173

Rn → R+ for which the hypotheses of Theorem 10.2.1 hold. In addition, letV be a positive function defined on R2n, i.e. V : R2n → R+ and let W be apositive matrix defined on R2n, W : R2n → R+ which satisfies the following:there is a positive number β so that W (xe) ≥ βI, ∀xe ∈ R2n.

Ewek{V [fe(xe) + ge(xe)s+Ge(xe)we

k]} − V (xe)

≤ Ewek{[fe(xe) + ge(xe)s+Ge(xe)we

k]TW (xe)[fe(xe) + ge(xe)s+Ge(x)we

k ]}

−xeTW (xe)xe.(10.27)

Let φe(xe) = γ2I−geT (xe)W (xe)ge(xe) and assume that there is a positive

number, a so that φe(xe)≥aI, ∀xe ∈ R2n. Suppose there is a matrix K(x)

of appropriate dimensions such that

feT (xe)[W (xe)ge(xe)φ−1e (xe)geT (xe)W (xe) +W (xe)]fe(xe) + heT (xe)he(xe)

−xeTW (xe)xe + Tr{GeT (xe)W (xe)Ge(xe)} ≤ 0, ∀xe ∈ R2n

(10.28)where he(xe) = u∗(x) − u∗(x).Then, the closed-loop system has an l2-gain≤ γ and is internally stable in themean square sense.

Proof: Let S(xe) = V (x) + V (xe). We shall show that S(xe) is a storagefunction for the system described by (10.20), together with the controller ofTheorem 10.2.1, and that this closed-loop system is dissipative with respectto the supply rate function γ2||v||2 − ||z||2 where z = col{h(x), D12u(x)}.Having shown this, it readily follows that closed-loop system has l2-gain≤ γ,that is

E{∞∑i=0

||zi||2} ≤ γ2E{||x0||2 +∞∑i=1

||vi||2}

for all stochastic processes {vi}i with finite energy, i.e.∑∞

i=1 E{||vi||2} <∞.

It is clear that because of the nature of V and V , S(xe) is a positivefunction and that S(0) = 0. To show now that the closed-loop system isdissipative in the sense mentioned above it is sufficient to show that there isa controller u such that

Ewk{S(fe(xe) + ge(xe)s+Ge(xe)wk)} − S(xe) + ||z||2 − γ2||v||2 ≤ 0,

∀xe ∈ R2n, v ∈ Rm

(10.29)which is equivalent to determining u so that

Ew1k{V (f(x) + g(x)u+ g1(x)v +G1(x)wk)} − V (x) + ||z||2 − γ2||v||2

+Ewek{V (fe(xe) + ge(xe)s+Ge(xe)we

k)} − V (xe) ≤ 0, ∀xe ∈ R2n, v ∈ Rm

174 10 H∞ Output-Feedback Control of Discrete-Time Systems

where we recall that s = v − v∗(x).Since, by assumption, there is a positive matrix Q(x) so that

Ew1k{V [f(x) + g(x)u + g1(x)v +G1(x)w1

k]} − V (x) ≤ −xTQ(x)x

+Ew1k{[f(x) + g(x)u+ g1(x)v +G1(x)w1

k ]TQ(x)[f(x) + g(x)u+ g1(x)v

+G1(x)w1k ]}

it is concluded from (10.22) thatEw1

k{V (f(x) + g(x)u+ g1(x)v +G1(x)wk)} − V (x) + ||z||2 − γ2||v||2

≤ [u− u∗]Tφu[u− u∗]− [v − v∗]Tφv[v − v∗].

Consider now the augmented system

xek+1 = fe(xek) + ge(xek)sk +G

e(xek)wek (10.30)

with the supply rate

γ2||s||2φv−||he(xe)||2φu

=γ2[v−v∗]Tφv[v−v∗]−[u−u∗]Tφu[u−u∗].

Using the BRL and (10.27), it follows that (10.28) renders the sufficient con-dition (10.29) for the system (10.30) to have l2-gain≤ γ. As to the stabilityof the closed-loop system, assuming that h(x) satisfies ||h(x)||2 ≥ α||x||2 forsome positive α, an application of Theorem 1.3.3 to (10.29) yields exponentialmean square stability of the closed-loop system. ��Assuming now f(x) = F (x), h(x) = H(x)x, and C(x) = C(x)x, we have thefollowing corollary which provides the desired sufficient condition for l2-gain≤γ in terms of nonlinear matrix inequalities rather than scalar inequalities.

Corollary 10.2.2 Assuming that all the hypotheses of Theorem 10.2.2 hold,together with above assumption regarding the structure of f, h and C. A suffi-cient condition for the closed-loop system to have l2=gain≤ γ is the following.

HT (x)H(x) + FT (x)[Q(x)− Q(x)g(x)φ−1u (x)gT (x)Q(x)]F (x) −Q

+rG1T (x)Q(x)G1(x) ≤ 0(10.31)

with the requirement that φv(x) > 0 for all x ∈ Rn and:

F eT(xe)[W (xe)ge(xe)φ−1e (xe)geT (xe)W (xe)+W (xe)]F e(xe)+HeT(xe)He(xe)

−W (xe) + [GeT (xe)W (xe)Ge(xe)] ≤ 0,

requiring

φe(xe) > 0, ∀xe ∈ R2n,

(10.32)where he(xe) = He(xe)xe = u∗(x)− u∗(x).

10.3 Conclusions 175

10.2.1 Example

We consider example 10.1.1 of the previous section, this time with partialinformation, that is, we assume the availability of x1 only. In fact, the mea-surements process is taken to be:

yk = Cxk +Dvk =[1 0

]xk + vk

We need to solve (10.31) and (10.32). Solving the first, for a prescribed γ, astate-feedback gain K(x) will be derived which is different from the constantgain found in Example 1 using Theorem 1. This feedback gain and the cor-responding matrix Q will be used in (10.28) to provide the filter gain K. Weobtained for γ = 2.55 that K(x) = −[11.1570 9.9434]F(x) and

Q =

[28.3112 12.380512.3805 11.0338

].

The corresponding filter gain was found to be K = [0.0097 0.2022]T with

W = 104

⎡⎢⎢⎣

6.2922 0.0202 −6.2098 −0.03530.0202 1.3329 0.0151 −1.3057−6.2098 0.0151 6.5412 0.1547−0.0353 −1.3057 0.1547 1.4127

⎤⎥⎥⎦.

Denoting Θ = I + g1φ−1v g1TQ we obtain

Θ =

[1 0

0.0008 1.0007

]

and the resulting output-feedback controller that renders closed-loop l2-gainof γ = 2.55 is described by uk = −[11.1570 9.9434]F(xk)xk where by (10.25)xk satisfies:

xk+1 = Θ[F (xk)+gK(xk)]xk+Kyk−K[C+Dφ−1v g1TQ(gK(xk)+F (xk))xk.

10.3 Conclusions

In this chapter we have developed some theory for H∞ output-feedbackcontroller design for nonlinear discrete-time stochastic systems. As a generaltheory for the discrete-time systems suffers from an inherent computationaldifficulties, we have resorted to a rather special case of quadratic storage func-tions which have facilitated the achievement of more complete results. Thus,we have established sufficient conditions for the underlying discrete-time non-linear stochastic system to have an L2-gain≤ γ subject to an output feedback.

176 10 H∞ Output-Feedback Control of Discrete-Time Systems

The resulting closed loop system has been shown to be exponentially meansquare stable, provided the associated storage function satisfies certain con-ditions.

An explicit formula for the output feedback controller has been derivedon the basis of two coupled HJI’s. Finally we have demonstrated the applica-bility of the theory by means of an example, where LMIs methods have beenemployed in solving the pertinent HJI’s.