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in many applications , the input and output of the controller are quantized signals . this is due to the physical properties of the actuators / sensors and the data - rate limitation of links connected to the controller . quantized control for linear time - invariant systems actively studied from various point of view , as surveyed in @xcite . moreover , in the context of systems with discrete jumps such as switched systems and piecewise affine ( pwa ) systems , control problems with limited information have recently received increasing attention . for sampled - data switched systems , a stability analysis under finite - level static quantization has been developed in @xcite , and an encoding and control strategy for stabilization has been proposed in the state feedback case @xcite , whose related works have been presented for the output feedback case @xcite and for the case with bounded disturbances @xcite . also , our previous work @xcite has studied the stabilization of continuous - time switched systems with quantized output feedback , based on the results in @xcite . however , relatively little work has been conducted on quantized control for pwa systems . in @xcite , a sufficient condition for input - to - state stability has been obtained for time - delay pwa systems with quantization signals , but logarithmic quantizers in @xcite have an infinite number of quantization levels .    the main objective of this paper is to stabilize discrete - time pwa systems with quantized signals . in order to achieve the local asymptotic stabilization of discrete - time pwa plants with finite data rates , we extend the event - based encoding method in @xcite . it is assumed that we are given feedback controllers that stabilize the closed - loop system in the sense that there exists a piecewise quadratic lyapunov function . in the input quantization case , the controller receives the original state . on the other hand , in the state quantization case , the quantized state and the currently active mode of the plant are available to the controller . the information on the active mode prevents a mode mismatch between the plant and the controller , and moreover , allows the controller side to recompute a better quantization value if the quantized state transmitted from the quantizer is near the boundaries of quantization regions . this recomputation is motivated in section 7.2 in @xcite . we also investigate the design of quantized feedback controllers . to this end , we consider the stabilization problem of discrete - time pwa systems with bounded disturbances ( under no quantization ) . the lypunov - based stability analysis and stabilization of discrete - time pwa systems has been studied in @xcite and @xcite in terms of linear matrix inequalities ( lmis ) and bilinear matrix inequalities ( bmis ) . in proofs that lyapunov functions decrease along the trajectories of pwa systems , the one - step reachable set , that is , the set to which the state belong in one step , plays an important role . in stability analysis , the one - step reachable set can be obtained by linear programming . by contrast , in the stabilization case , since the next - step state depends on the control input , it is generally difficult to obtain the one - step reachable set . therefore many previous works for the design of stabilizing controllers assume that the one - step reachable set is the total state space . however , if disturbances are bounded , then this assumption leads to conservative results and high computational loads as the number of the plant mode increases . we aim to find the one - step reachable set for pwa systems with bounded disturbances . to this effect , we derive a sufficient condition on feedback controllers for the state to belong to a given polyhedron in one step . this condition can be used to add constraints on the state and the input as well . furthermore , we obtain a set containing the one - step reachable set by using the information of the input matrix @xmath0 and the input bound @xmath1 . this set is conservative because the affine feedback structure @xmath2 for mode @xmath3 is not considered , but it can be used when we design the polyhedra that are assumed to be given in the above sufficient condition . combining the proposed condition with results in @xcite for lyapunov functions to be positive and decrease along the trajectories , we can design stabilizing controllers for pwa systems with bounded disturbances . this paper is organized as follows . the next section shows a class of quantizer and a basic assumption on stability . in sections iii and iv , we present an encoding strategy to achieve local stability for pwa systems in the input quantization case and the state quantization case , respectively . in section v , we study the one - step reachable set for the stabilization problem of pwa systems with bounded disturbances . finally , concluding remarks are given in section vi .    due to space constraints , all proofs and a numerical example have been omitted and can be found in * ? ? ? _ notation _ : for a set @xmath4 , we denote by @xmath5 the closure of @xmath6 . for sets @xmath7 , let @xmath8 denote their minkowski sum . let @xmath9 and @xmath10 denote the smallest and the largest eigenvalue of @xmath11 . let @xmath12 denote the transpose of @xmath13 . for @xmath14 , we denote the @xmath15-th entry of @xmath16 by @xmath17 . let @xmath18 be a vector all of whose entries are one . for vectors @xmath19 , the inequality @xmath20 means that @xmath21 for every @xmath22 . on the other hand , for a square matrix @xmath23 , the notation @xmath24 ( @xmath25 ) means that @xmath26 is symmetric and semi - positive ( positive ) definite .    the euclidean norm of @xmath14 is denoted by @xmath27 . the euclidean induced norm of @xmath13 is defined by @xmath28 . the @xmath29-norm of @xmath30^{\top}$ ] is denoted by @xmath31 , and the induced norm of @xmath13 corresponding to the @xmath29-norm is defined by @xmath32 . for @xmath33 , let @xmath34 and @xmath35 . we consider the following class of discrete - time pwa systems : @xmath36 where @xmath37 is the state and @xmath38 is the control input . the set @xmath39 is divided into finitely many disjoint polyhedra @xmath40 : @xmath41 . we denote the index set @xmath42 by @xmath43 .    given a feedback gain @xmath44 and an affine term @xmath45 for each mode @xmath46 , the control input is in the affine state feedback form : @xmath47 we assume that @xmath48 if @xmath49 . we will study the design of @xmath50 and @xmath51 in section v , but for quantized control in sections iii and iv , @xmath50 and @xmath51 are assumed to be given .      in this paper , we use the class of quantizers proposed in @xcite . let @xmath52 be a set composed of finitely many points in @xmath53 . a quantizer @xmath54 is a piecewise constant function from @xmath55 to @xmath52 . geometrically , this means that @xmath56 is divided into a finite number of quantization regions of the form @xmath57 @xmath58 . for the quantizer @xmath54 , we assume that there exist @xmath59 with @xmath60 such that @xmath61 : @xmath62 the quantized value @xmath63 is the data on @xmath64 transmitted to the controller at time @xmath65 . we adjust @xmath66 to obtain detailed information on @xmath64 near the origin . define @xmath67 which is the one - step reachable set from @xmath68 for the pwa system and the state feedback law without quantization . define also @xmath69 we assume that the following stability of the closed - loop system is guaranteed by a piecewise quadratic lyapunov function :    [ ass : lyapunov_for_qs ] _ consider the pwa system with given affine feedback . define a function @xmath70 by @xmath71 where @xmath72 and @xmath73 are symmetric matrices . there exist @xmath74 and @xmath75 for @xmath76 , such that the lypunov function @xmath77 defined by @xmath78 @xmath79 satisfies @xmath80 for every @xmath81 , @xmath82 , and @xmath83 , where @xmath84 is defined by @xmath85 _    in section v , we will discuss how to obtain @xmath84 of in the design process of @xmath50 and @xmath51 . in this section , we study stabilization with quantized input : @xmath86 the closed - loop system we consider is given by @xmath87 we place the following assumption on the state transition :    [ ass : input_ver ] _ define @xmath88 . for every @xmath81 , the one - step reachable set @xmath89 in satisfies @xmath90 . _    the condition @xmath91 implies that @xmath92 is invariant for the system , and checking this condition is closely related to how to derive @xmath84 in . in section v , we will derive sufficient conditions on @xmath50 and @xmath51 for @xmath93 to hold ; see remark [ rem : total_ss_cond ] . first we fix the zoom parameter @xmath94 . similarly to @xcite , we show that the lyapunov function decreases until the state gets to the corresponding level set . [ thm : mktom ] _ consider the pwa system with given @xmath50 and @xmath51 . let assumptions [ ass : lyapunov_for_qs ] and [ ass : input_ver ] hold . fix @xmath95 , and define @xmath96 also , let @xmath97 for all @xmath81 , and set @xmath98 define @xmath99 and @xmath100 by @xmath101 if @xmath102 and @xmath103 satisfy @xmath104 then all solutions of that start in @xmath105 enter @xmath100 in a finite time @xmath106 satisfying @xmath107 furthermore , if @xmath108 holds , then the solution @xmath109 belongs to @xmath105 for all @xmath110 . _    in order to utilize , first we show that if @xmath111 , then there exists @xmath112 such that @xmath113 . suppose that @xmath111 . define @xmath114 by @xmath115 . then we have @xmath116 . since @xmath117 , it follows that @xmath118 . moreover , @xmath119 implies @xmath120 , and hence @xmath121 from and @xmath122 . we therefore obtain @xmath123 therefore , from assumption [ ass : input_ver ] , there exists @xmath124 such that @xmath125 combining and , we have @xmath126 . thus @xmath82 by definition .    in what follows , for simplicity of notation , we omit the indices @xmath3 and @xmath127 of @xmath128 , @xmath129 , @xmath130 , @xmath131 , and @xmath132 . define @xmath133 and @xmath134 . since holds , we have @xmath135      & \qquad \quad+ 2|\bar a_i^{\top}q_j e_k| \cdot |x_k| +      e_k^{\top}q_je_k + 2 h_{ij } ^{\top } e_k \\      & =      -\varepsilon \gamma |x_k|^2      - ( 1\!-\ ! \varepsilon ) ( 1\!-\!\delta ) \gamma       \left (       |x_k| \!-\ ! \frac{|\bar a_i^{\top}q_j e_k|}{(1\!-\!\varepsilon ) ( 1\!-\!\delta ) \gamma }       \right)^2 \\[-2pt ]      & \qquad \quad - ( 1\!-\ ! \varepsilon ) \delta \gamma |x_k|^2       +      e_k^{\top}q_je_k        + \frac{|\bar a_i^{\top}q_j e_k|^2 } { ( 1\!-\ ! \varepsilon ) ( 1\!-\ ! \delta ) \gamma }      + 2h_{ij}^{\top}e_k \\      & \leq      -\varepsilon \gamma |x_k|^2 - \upsilon ,                                              \end{aligned}\ ] ] where @xmath136 if @xmath119 , then @xmath137 hence , noticing @xmath138 , we have @xmath139 therefore @xmath140 for every @xmath81 , we obtain @xmath141 whenever @xmath142 . note that the most right side of is independent of the plant mode @xmath3 .    by , we have @xmath143 in conjunction with , this shows that if the initial state @xmath144 belongs to @xmath99 , then @xmath145 holds for some integer @xmath106 satisfying . let us next prove that @xmath146 is an invariant region for the system . from , @xmath147 until @xmath148 . once @xmath149 , we have @xmath150 therefore if holds , then @xmath151 leads to @xmath152 . the state trajectories again go to @xmath153 while belonging to @xmath99 since assumption [ ass : input_ver ] gives @xmath154 for all @xmath110 , we see that @xmath155 for all @xmath110 .    as in @xcite , we can achieve the state convergence to the origin by adjusting the zoom parameter @xmath156 :    [ thm : input_convergence ] _ consider the pwa system with given @xmath50 and @xmath51 . let assumptions [ ass : lyapunov_for_qs ] and [ ass : input_ver ] hold . let the initial state @xmath157 and the initial zoom parameter @xmath158 . assume that and hold , and define @xmath159 adjust @xmath156 by @xmath160 when @xmath109 gets to @xmath161 , and send to the plant the quantized input @xmath162 at time @xmath65 if @xmath117 . this event - based update strategy of @xmath156 leads to @xmath163 . _ first we prove that as long as the quantizer does not saturate , the state trajectory belongs to @xmath39 and the holds for all @xmath110 . define @xmath164 } : =      \ { b_i d : |d|_{\infty } \leq \omega^p \delta      \}$ ] . since @xmath164 } \subset \mathcal{b}_i$ ] , if assumption [ ass : input_ver ] holds , then @xmath165 }      \subset { \bf x }      $ ] @xmath166 for every @xmath167 . hence @xmath154 for all @xmath110 unless the quantizer saturates . moreover , if we define @xmath168}$ ] by the set consisting of all @xmath124 satisfying @xmath169 as in , then @xmath168 } \subset \mathcal{s}_i^{[0 ] } = \mathcal{s}_i$ ] . thus holds for every @xmath81 and @xmath170}$ ] , and hence we have for all @xmath110 unless the quantizer saturation occurs . let an update occur at @xmath171 , i.e. , @xmath172 and @xmath173 . then we have @xmath174 therefore @xmath175 defined by @xmath176 satisfies @xmath177 . since @xmath172 , it follows that @xmath178 . hence theorem [ thm : mktom ] shows that for all @xmath110 , @xmath179 , which means @xmath180 and the quantizer does not saturate for every @xmath110 . moreover , the update period does not exceed @xmath181 in . since @xmath182 , it follows that @xmath183 as @xmath184 . thus @xmath185 as @xmath184 .    for _ continuous - time _ systems , the level sets @xmath99 and @xmath100 are invariant regions of the state trajectories  @xcite . however , for _ discrete - time _ systems , @xmath100 may not be invariant . we therefore need the event - based adjustment of the zoom parameter as in ( * ? ? ? * section iii ) and @xcite . let us next study stabilization of pwa systems with quantized state feedback . we assume that the controller receives the information on the quantized state and the active mode . [ ass : con_qs_am ]    under assumption [ ass : con_qs_am ] , the control @xmath186 is given by @xmath187 the closed - loop system we consider can be written in this way : @xmath188      we place an assumption similar to assumption [ ass : input_ver ] . [ ass : state_ver ] _ define @xmath189 . for every @xmath81 , the one - step reachable set @xmath89 in satisfies @xmath90 . _    see remark [ rem : total_ss_cond ] for the condition @xmath190 .    as in the input quantization case , we first fix @xmath191 and obtain a result similar to theorem [ thm : mktom ] , based on the technique in @xcite . [ thm : mtom ] _ consider the pwa system with given @xmath50 and @xmath51 . let assumptions [ ass : lyapunov_for_qs ] and [ ass : state_ver ] hold . fix @xmath95 . define @xmath192 and @xmath193 as in and respectively , and define @xmath131 and @xmath132 by @xmath194 set @xmath195 , @xmath196 , @xmath102 , @xmath197 , and @xmath198 as in theorem [ thm : mktom ] , and set @xmath199 define @xmath200 and @xmath201 by @xmath202 if @xmath203 satisfies @xmath204 then all solutions of that start in @xmath205 enter @xmath201 in a finite time @xmath106 satisfying @xmath206 and @xmath207 can be observed from @xmath208 . furthermore , if @xmath209 then the solution belongs to @xmath205 for all @xmath110 . _    if we define @xmath210 , then the proof follows the same lines as that of theorem 3.7 until . we see that the lyapunov function decreases if the initial state belongs to @xmath205 and if the state does not arrive at @xmath153 . we show that the quantized state @xmath211 gets to @xmath212 at @xmath213 as follows . suppose , on the contrary , that @xmath214 for all @xmath215 . if @xmath216 , we have @xmath217 from @xmath218 . therefore we have @xmath219 for all @xmath220 . however , if @xmath219 for all @xmath221 , then the lyapunov function decreases as , and hence @xmath222 . this implies that @xmath223 , which leads to a contradiction .    from @xmath217 , we observe that @xmath224 and hence that @xmath225 . [ fig : state_trajectory ] illustrates the regions used in this proof . the invariance of @xmath205 for the state trajectories can be proved as in theorem [ thm : input_convergence ] . this completes the proof .    in the input quantization case of theorem [ thm : input_convergence ] , we use the _ original _ state for the adjustment of the zoom parameter @xmath156 . by contrast , in the state quantization case , we can achieve the asymptotic stability by adjusting @xmath156 with the _ quantized _ state . [ thm : state_convergence ] _ consider the pwa system with given @xmath50 and @xmath51 . let assumptions [ ass : lyapunov_for_qs ] and [ ass : state_ver ] hold . let the initial state @xmath226 and the initial zoom parameter @xmath158 . assume that and hold , and define @xmath227 adjust @xmath156 by @xmath160 when @xmath228 gets to @xmath229 , where @xmath230 , and send to the controller the quantized state @xmath231 at time @xmath65 . this event - based update strategy of @xmath156 leads to @xmath163 . _ if we observe @xmath232 at time @xmath65 , then @xmath233 , where @xmath234 . hence we obtain @xmath235 after the update @xmath236 . the other part of the proof follows in the same line as that of theorem [ thm : input_convergence ] , so we omit it . another approach to stabilize the pwa system with the quantized state feedback is to combine the plant and the quantizer . in this case , we consider the following pwa system : @xmath237 the difficulty of this approach is that we need to stabilize pwa systems with output feedback @xmath238 . output feedback stabilization of pwa systems has been studied in @xcite and the reference therein , but the output structure in these previous works is @xmath239 . in general , it is difficult to design stabilizing controllers for the system . moreover , if we adjust the quantizer , then the system becomes time varying . to avoid technical issues , we do not proceed along this lines .      as in ( * section 7.2 ) , a better quantization value can be computed in the controller side if the state is near switching boundaries . for the recompution of a new quantization value , we make the following assumption :    [ ass : qrp ] _ the controller has the information on the switching regions @xmath240 . all quantization regions @xmath241 are polyhedra . _    if the quantized state @xmath211 is in a quantization region that has no switching boundary , then the controller uses @xmath211 . on the other hand , in order to achieve better performance , if the corresponding quantization region contains a switching boundary , then the controller can generate a new quantized value from the information on the quantized state and the currently active mode as follows .    let the switching region corresponding to the active mode be @xmath68 and let the quantization region of the transmitted quantized state be @xmath241 . then the state belongs to @xmath242 . suppose that @xmath242 is bounded . otherwise , the controller does not recompute a new quantization value . since both regions are polyhedra , @xmath242 is a polyhedron . let us denote its closure by @xmath243 .    since @xmath244 , the controller computes a new quantized state latexmath:[\[q_{\text{new } } : =   { \mathop{\rm argmin}}_{\xi \in \mathbb{r}^{\sf n } } \max_{x \in \mathcal{a } }    @xmath243 . the next theorem shows that @xmath246 can be obtained by linear programming and that the quantization error by using @xmath246 as the new quantized state is always less than or equal to the quantization level @xmath247 in . [ thm : new_quntize ] _ let the vertices of @xmath243 be @xmath248 . the new quantization value @xmath249 is computed by the following linear program : @xmath250 moreover , if @xmath251 , then @xmath249 satisfies @xmath252 _    it is well known that for every @xmath253 , @xmath254 ; see also appendix . hence the linear program gives @xmath246 . since @xmath255 , it follows from that if @xmath256 , then @xmath257 where @xmath258 is the original quantization value of @xmath241 .    * ( a ) * if the original quantization region @xmath241 is a polyhedron , then the zoomed - in quantization region @xmath259 is also a polyhedron . we can therefore compute the new quantization value @xmath246 after adjusting the zoom parameter @xmath156 as well .    * ( b ) * the use of @xmath246 does not affect the stability analysis in theorems [ thm : mtom ] and [ thm : state_convergence ] , because its quantization error does not exceed @xmath247 . to obtain @xmath246 , we need to solve the linear program . if the computation is not finished by the time when the control input is generated , then the controller can use the original quantization value @xmath258 . for quantized control , here we aim to find a feedback gain @xmath50 and an affine term @xmath51 satisfying and for every @xmath81 , @xmath82 , and @xmath83 . to this effect , we show how to obtain a set containing @xmath260 in with less conservatism . let us consider discrete - time pwa systems with affine state feedback control under no quantization . theorem 1 in @xcite shows that in order to stabilize the pwa system , it is enough to find a feedback gain @xmath50 and an affine term @xmath51 for every @xmath81 such that @xmath261 ( @xmath83 ) and the piecewise lyapunov function @xmath262 satisfies and @xmath263 for some @xmath264 . define @xmath78 ( @xmath83 ) , with a function @xmath265 . the sufficient condition of used for the stability analysis in @xcite is that @xmath266 for all @xmath83 and @xmath124 with @xmath267 , where @xmath89 is the one - step reachable set defined in . however , it is generally difficult to obtain @xmath50 and @xmath51 satisfying this condition in a less conservative way . this is because @xmath127 , namely , the polyhedron to which @xmath268 may belong is dependent of the unknown variables @xmath50 , @xmath51 . to circumvent this difficulty , it is assumed , e.g. , in @xcite that the state can reach every polyhedron in one step , but this assumption makes the controller synthesis conservative if disturbances are bounded . in addition to that , checking the condition for every pair @xmath269 leads to computational complexity for pwa systems with large number of modes . therefore the objective here is to obtain a set to which the state go in one step under bounded disturbance .      consider a pwa system with bounded disturbances given by @xmath270 where the disturbance @xmath114 satisfies @xmath271 for all @xmath110 . the next lemma gives a motivation of studying the set @xmath84 defined in in terms of practical input - state - stability in addition to quantized control in the previous sections . a proof is provided for completeness . _ let @xmath272 . define @xmath273 and @xmath274 . for every @xmath81 , assume that @xmath275 . if the piecewise lyapunov function @xmath78 ( @xmath83 ) , with a function @xmath265 , satisfies for some @xmath276 and there exist @xmath277 and @xmath278 such that for every @xmath81 and @xmath82 and for every @xmath83 and @xmath279 , @xmath280 then we have @xmath281 where @xmath282 . _    since @xmath283 for all @xmath117 , it follows that if @xmath117 , then @xmath113 for some @xmath82 . therefore and give @xmath284 and hence @xmath285 using again , we obtain from . first we study the case when @xmath50 and @xmath51 are known . the lemma below gives a condition equivalent to @xmath286 in the definition of @xmath84 . [ lem : min_sum_trans ] _ define @xmath287 . for arbitrary sets @xmath288 , we have @xmath289 _    it suffices to show that if there exists @xmath290 satisfying @xmath291 , then we have @xmath292 such that @xmath293    since @xmath294 , it follows that @xmath295 for some @xmath296 and for some @xmath279 , and also that @xmath297 . moreover , since @xmath298 , we have @xmath299 the desired conclusion holds with @xmath300 . we see from lemma [ lem : min_sum_trans ] that @xmath301 is equivalent to @xmath302 . therefore @xmath84 in satisfies @xmath303    the following theorem gives a set containing @xmath84 , which can be obtained by linear programing :    [ lem : set_containing_si ] _ using suitable @xmath304 and @xmath305 , we can write the closure of @xmath68 as @xmath306 define @xmath84 as in . if we define @xmath307 by @xmath308 then @xmath309 . _    first of all , we see that there exists @xmath310 satisfying both @xmath311 and @xmath312 if and only if there exists @xmath83 such that @xmath313 , where @xmath314 and @xmath315 .    by definition , @xmath316 is equivalent to @xmath317 for some @xmath318 and @xmath319 satisfying @xmath320 and @xmath321 . therefore @xmath316 is equivalent to @xmath322 for some @xmath319 . thus we obtain the following fact : if @xmath323 , then @xmath324 noticing that @xmath124 satisfies if and only if @xmath325 , we have that @xmath309 . the conservatism of theorem [ lem : set_containing_si ] is due to only @xmath326 . if we allow more conservative results , then we can use the set @xmath327 below , which can be obtained with less computational cost by removing the disturbance term @xmath328 . a similar idea is used for the analysis of reachability with bounded disturbance in @xcite . [ cor : tilde_sei ] _ let @xmath329 be the sum of the absolute value of the elements in @xmath15-th row of @xmath330 and define @xmath331^{\top}$ ] , where @xmath332 is the number of rows of @xmath333 . if we define @xmath334 by @xmath335 then @xmath84 in satisfies @xmath336 . _    it suffices to prove that @xmath337 indeed , if holds , then @xmath323 implies @xmath338 where @xmath339 and @xmath340 . this leads to @xmath336 . let us study the first element of @xmath341 . let @xmath342 , @xmath343 , and @xmath344 be the @xmath345-th entry of @xmath346 , @xmath333 and the first entry of @xmath347 , respectively . also let @xmath348 and @xmath349 be the @xmath15-th element of @xmath350 and @xmath328 , respectively . if @xmath351 and @xmath279 , then the first element @xmath352 of @xmath341 satisfies @xmath353 since we have the same result for the other elements of @xmath341 , it follows that holds . let us next investigate the case when @xmath50 and @xmath51 are unknown . the set @xmath307 given in theorem [ lem : set_containing_si ] works for stability analysis in the presence of bounded disturbances , but @xmath307 is dependent on the feedback gain @xmath50 and the affine term @xmath51 . hence we can not use it for their design . here we obtain a set @xmath354 , which does not depend on @xmath50 , @xmath51 . moreover , we derive a sufficient condition on @xmath50 , @xmath51 for the state to belong to a given polyhedron in one step . let @xmath355 be the polyhedron defined by @xmath356 and we make an additional constraint that @xmath357 for all @xmath110 . similarly to @xcite , using the information on the input matrices @xmath0 and the input bound @xmath355 , we obtain a set independent of @xmath50 , @xmath51 to which the state belong in one step . [ coro : si_sufficient_with_disturbance ] _ assume that for each @xmath81 , @xmath358 and @xmath45 satisfy @xmath359 and @xmath360 if @xmath83 and @xmath279 . let the closure of @xmath68 be given by . define @xmath361 then we have @xmath362 for all @xmath83 and @xmath279 , and hence @xmath84 in satisfies @xmath363 . _ define @xmath364 . to show , it suffices to prove that for all @xmath83 and @xmath279 , there exists @xmath365 such that @xmath366 . suppose , on the contrary , that there exist @xmath83 and @xmath279 such that @xmath367 for every @xmath365 . since @xmath368 , it follows that @xmath369 for some @xmath124 . also , by definition @xmath370 since @xmath83 , @xmath371 , @xmath279 , and @xmath372 , it follows that @xmath365 . hence we have @xmath372 for some @xmath365 . thus we have a contradiction and holds for every @xmath83 and @xmath279 . let us next prove @xmath363 . let @xmath82 . by definition , there exists @xmath83 and @xmath279 such that @xmath373 . also , we see from that there exists @xmath374 such that @xmath375 . hence we have @xmath376 , which implies @xmath377 . thus we have @xmath363 . see remark [ rem : total_ss_cond ] for the assumption that @xmath378 and @xmath379 for all @xmath380 .    * ( a ) * in theorem [ coro : si_sufficient_with_disturbance ] , we have used the counterpart of @xmath307 given in theorem [ lem : set_containing_si ] , but one can easily modify the theorem based on @xmath334 in corollary [ cor : tilde_sei ] .    * ( b ) * if @xmath0 is full row rank , then for all @xmath381 , @xmath382 , and @xmath279 , there exists @xmath383 such that @xmath384 . in this case , we have the trivial fact : @xmath385 . theorem [ coro : si_sufficient_with_disturbance ] ignores the affine feedback structure @xmath386 ( @xmath83 ) , which makes this theorem conservative . since the one - step reachable set depends on the unknown parameters @xmath50 and @xmath51 , we can not utilize the feedback structure unless we add some conditions on @xmath50 and @xmath51 . in the next theorem , we derive linear programming on @xmath50 and @xmath51 for a bounded @xmath68 , which is a sufficient condition for the one - step reachable set under bounded disturbances to be contained in a given polyhedron . [ thm : add_cond_ki ] _ let a polyhedron @xmath387 , and let @xmath68 be a bounded polyhedron . let @xmath388 and @xmath389 be the vertices of @xmath390 and @xmath391 , respectively . a matrix @xmath358 and a vector @xmath45 satisfy @xmath392 for all @xmath83 and @xmath393 if linear programming @xmath394 is feasible for every @xmath395 and for every @xmath396 . _    define @xmath364 . relying on the results ( * ? ? ? * chap . 6 ) ( see also @xcite ) , we have @xmath397 where @xmath398 means the convex hull of a set @xmath399 . we therefore obtain @xmath400 for all @xmath381 and @xmath393 if and only if @xmath401 , or , holds for every @xmath395 and @xmath402 . thus the desired conclusion is derived .    * ( a ) * to use theorem [ thm : add_cond_ki ] , we must design a polyhedron @xmath403 in advance . one design guideline is to take @xmath403 such that @xmath404 for some @xmath405 , where @xmath406 is defined in theorem [ coro : si_sufficient_with_disturbance ] .    * ( b ) * as in theorem [ lem : set_containing_si ] , the conservatism in theorem [ thm : add_cond_ki ] arises only from @xmath407 .    * ( c ) * theorem [ thm : add_cond_ki ] gives a trade - off on computational complexity : in order to reduce the number of pairs such that holds , we need to solve the linear programming problem .    * ( d ) * when the state is quantized , then @xmath408 in , and hence @xmath409 depends on @xmath50 linearly . in this case , however , theorem [ thm : add_cond_ki ] can be used for the controller design . [ rem : total_ss_cond ] assumptions [ ass : input_ver ] , [ ass : state_ver ] and theorem [ coro : si_sufficient_with_disturbance ] require conditions on @xmath50 and @xmath51 that @xmath410 and @xmath360 for all @xmath83 and all @xmath279 . if @xmath411 and @xmath412 , then these conditions always hold . if @xmath413 but if @xmath68 is a bounded polyhedron , then theorem [ thm : add_cond_ki ] gives linear programming that is sufficient for @xmath410 to hold . also , theorem [ thm : add_cond_ki ] with @xmath414 , @xmath415 , and @xmath416 can be applied to @xmath360 . if @xmath410 and @xmath360 hold for bounded @xmath417 and @xmath355 , then we can easily set the quantization parameter @xmath203 in to avoid quantizer saturation . similarly , we can use theorem [ thm : add_cond_ki ] for constraints on the state and the input .    by theorems [ coro : si_sufficient_with_disturbance ] and [ thm : add_cond_ki ] , we obtain linear programing on @xmath50 and @xmath51 for a set containing the one - step reachable set under bounded disturbances . however , in lmi conditions of @xcite for and , @xmath50 is obtained via the variable transformation @xmath418 , where @xmath419 and @xmath420 are auxiliary variables . without variable transformation / elimination , we obtain only bmi conditions for to hold as in theorem 7.2.2 of @xcite . the following theorem also gives bmi conditions on @xmath50 for and to hold , but we can apply the cone complementary linearization ( ccl ) algorithm  @xcite to these bmi conditions :    [ thm : pwa_lmi ]    _ consider the pwa system with control affine term @xmath421 . let a matrix @xmath422 satisfy @xmath423 if @xmath416 and @xmath424 and if there exist @xmath425 , @xmath50 , and @xmath426 with all elements non - negative such that @xmath427 and @xmath428 hold for all @xmath76 and @xmath82 , then there exist @xmath429 such that @xmath430 ( @xmath83 ) satisfies and for every @xmath76 , @xmath82 , and @xmath83 . _    furthermore , consider the case @xmath431 and @xmath432 . for given @xmath433 with @xmath434 , if there exist @xmath425 , @xmath50 , and @xmath426 with all elements non - negative such that @xmath435 and @xmath428 hold for all @xmath76 and @xmath82 , then there exist @xmath436 such that @xmath430 ( @xmath83 ) satisfies and for every @xmath76 , @xmath82 , and @xmath83 . since the positive definiteness of @xmath437 implies , it is enough to show that and lead to and , respectively .    for @xmath438 satisfying the second lmi in and , we have @xmath439 . furthermore , @xmath440 if and only if @xmath441 . define @xmath442 . applying the schur complement formula to the lmi condition in , we have @xmath443 since @xmath444 , there exists @xmath75 such that @xmath445 for every @xmath83 . hence we obtain .    as regards , it follows from theorem 3.1 in @xcite that holds for some @xmath446 if @xmath447 pre- and post - multiplying @xmath448 and using the schur complement formula , we obtain the first lmi in . since @xmath449 , the conditions in theorem [ thm : pwa_lmi ] are feasible if the problem of minimizing @xmath450 under / has a solution @xmath451 . in addition to lmis and , we can consider linear programming for the constraint on the one - step reachable set . the ccl algorithm solves this constrained minimization problem . the ccl algorithm may not find the global optimal solution , but , in general , we can solve the minimization problem in a more computationally efficient way than the original non - convex feasibility problem  @xcite . consider a pwa system in with quantized state feedback , where @xmath452 the matrix @xmath304 and the vector @xmath305 in characterizing the region @xmath68 are given by @xmath453 let @xmath454 and let us use a uniform - type quantizer whose parameters in are @xmath455 and @xmath456 . by using theorems [ thm : add_cond_ki ] and [ thm : pwa_lmi ] , we designed feedback gains @xmath50 such that the lyapunov function @xmath430 ( @xmath83 ) satisfies and for every @xmath76 , @xmath82 , and @xmath83 , and the following constraint conditions hold @xmath457 the resulting @xmath50 were given by @xmath458 and we obtained the decease rate @xmath459 in of the `` zoom '' parameter @xmath156 with @xmath460 and @xmath461 . [ fig : state_trajectory_example ] shows the state trajectories with initial states on the boundaries @xmath462 and @xmath463 . we observe that all trajectories converges to the origin and that the constraint conditions and are satisfied in the presence of quantization errors . we have provided an encoding strategy for the stabilization of pwa systems with quantized signals . for the stability of the closed - loop system , we have shown that the piecewise quadratic lyapunov function decreases in the presence of quantization errors . for the design of quantized feedback controllers , we have also studied the stabilization problem of pwa systems with bounded disturbances . in order to reduce the conservatism and the computational cost of controller designs , we have investigated the one - step reachable set .    here we give the proof of the following proposition for completeness :     _ let @xmath464 be a bounded and closed polyhedron , and let @xmath248 be the vertices of @xmath243 . for every @xmath465 , we have @xmath466 _    choose @xmath244 arbitrarily , and let @xmath467 let @xmath468-th entry of @xmath469 , @xmath350 , and @xmath470 be @xmath471 , @xmath472 , and @xmath473 , respectively . for every @xmath468 , we have @xmath474 hence @xmath475 this completes the proof . the first author would like to thank dr . k. okano of university california , santa barbara for helpful discussions on quantized control for pwa systems . the authors are also grateful to anonymous reviewers whose comments greatly improved this paper . m.  xiaowu and g.  yang , `` global input - to - state stabilization with quantized feedback for discrete - time piecewise affine systems with time delays , '' _ j. syst . sci . complexity _ , vol . 26 , pp . 925939 , 2013 . j.  qiu , g.  feng , and h.  gao , `` approaches to robust @xmath477 static output feedback control of discrete - time piecewise - affine systems with norm - bounded uncertainties , '' _ int . j. robust and nonlinear control _ , vol . 21 , pp . 790814 , 2011 . l.  e. ghaoui , f.  oustry , and m.  aitrami , `` a cone complementarity linearization algorithm for static output - feedback and related problems , '' _ ieee trans . 11711176 , 1997 .
this paper studies quantized control for discrete - time piecewise affine systems . for given stabilizing feedback controllers 
 , we propose an encoding strategy for local stability . 
 if the quantized state is near the boundaries of quantization regions , then the controller can recompute a better quantization value . 
 for the design of quantized feedback controllers , we also consider the stabilization of piecewise affine systems with bounded disturbances . in order to derive a less conservative design method with low computational cost 
 , we investigate a region to which the state belong in the next step .