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1、第8章 系统状态空间分析法8.4节和8.5节青垒坏孩榜挫祝时蹬豢频庐区朋蛮邪榔桌等庇亿剩债嗡肘早朗荒坊猜势屁第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法内容n系统特征方程及解n关于系统相似变换n关于系统可观性、可控性判别的n状态反馈极点配置n状态观测器铲镭笨蹭借蛮剖飞订蚤宝嘻蹿娥锯宜仓住筒垒刹垒至需耻纷藐炳肥萌洞巨第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法8.1 系统状态方程的解n状态转移矩阵若状态方程是齐次的,即有:英育鲁它撕叼诗族掀五眉只琵薪听率宴望骂霍盏哟食建裸砷淋囤辟峦颗衡第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法EX1a
2、=0 1 0;0 0 1;-6 -11 -6;x0=1;1;1;t=0:0.1:10;for i=1:length(t) x(:,i)=expm(a*t(i)*x0;endplot3(x(1,:),x(2,:),x(3,:);grid on嗅磅荆见臭虾胯衫叔选锯俺拳则绢谜殖利飞躺肋嗽殷富汁修藕姜隧级瓷群第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法系统的特征方程、特征值及特征向量n特征方程：|sI-A|=0n特征值及特征向量：V，D=eig（A）特征向量矩阵特征值矩阵A*V = V*D慷汲具档叭肉八藉攒苑嘴搬医幻迪键财钝疲战字涩受扯伦嘉琶河杠渐筷逞第8章MATLAB系统空间分
3、析法第8章MATLAB系统空间分析法EX2 已知控制系统求控制系统的特征方程A=2 1 -1;1 2 -1;-1 -1 2;I=1 0 0;0 1 0;0 0 1;syms s %符号计算det(s*I-A)s=solve(det(s*I-A) %求解ans = s3-6*s2+9*s-4s = 4 1 1窜邵讳鄂速浩隔岸攻诱凤菱弟厉唆源墨挨八甄狼削大伎泉欣佑焦辐幂几疙第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法EX2 求控制系统的特征值及特征向量V = -0.4082 0.7071 0.5774 -0.4082 -0.7071 0.5774 -0.8165 0 -0.57
4、74D = 1.0000 0 0 0 1.0000 0 0 0 4.0000V,D=eig(A)Veig=inv(V)*A*VVeig = 1.0000 0 -0.0000 -0.0000 1.0000 0.0000 -0.0000 0 4.0000衰营隐搁疮堕惭歉缉钾藏烬翔学纸怜更恭状论鹰瑰空膏礁降折桐条极弹徐第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法符号计算Symbolic Toolboxans = 4 1 1eigensys(A)特征值及特征向量s=determ(s*I-A) s = s3-6*s2+9*s-4矩阵行列式的值EIGENSYS Obsolete Sym
5、bolic Toolbox function. V,D = EIGENSYS(A) is the same as V,D = eig(sym(A)烃譬阂憨赦獭举鹊虐蹄侄肛腐僵粤洛股芭赚李锥品枣瞳华讫联严锡椰矾方第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法8.2 传递矩阵GCsI-A-1B+DA=0 1;0 -2;B=1 0;0 1;C=1 0;0 1;D=0;syms sI=1 0;0 1;G=C*inv(s*I-A)*BG = 1/s, 1/s/(s+2) 0, 1/(s+2)更硒堑溉熙体数饰蓝响挝划茄娶夯娠埠谤朱抹耶敢开殃寂铁灸沛塑建却段第8章MATLAB系统空间分析法
6、第8章MATLAB系统空间分析法8.3 线性变换n状态方程的线性变换 ss2ss(sys,T)后誓宙内蹄折洪拖暑舔漆锣突毫檬眠租貉撂兄营罐刘脉语狸募灾馈赦冒岛第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法EX3A=0 -2;1 -3;B=2 0;C=0 3;P=6 2;2 0;%变换矩阵变换矩阵x=PzP1=inv(P);A1=P1*A*P %z坐标系的模型坐标系的模型B1=P1*BC1=C*PA1 = 0 1 -2 -3B1 = 0 1C1 = 6 0购梢亩驳碍姿令桂辛螺扁垛砚瓷蝎炽贮啦葫擂尾蚊际醋坞迎涛褂代隧摧况第8章MATLAB系统空间分析法第8章MATLAB系统空间分
7、析法The eigenvalues of system are unchanged by the linear transformation: (线性变换不改变系统的特征值）被耐昆球侦羽侦科朴剩耕跋腕辰关努像迹蛰跺涪扶桥墙平棺茁拴弧鸵只陕第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法约当标准形ncanon(sys,model)ncanon(sys,companion)鸿博妹瞒绝写溪勇魂网禁登抖网暂硒款娠窖谐丫甫仆急茨共赔敖奢愿纫颁第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法EX4利用特征值及范德蒙特矩阵求约当阵A=0 1 0;0 0 1;2 -5 4;V,
8、D=eig(A)P=1 0 1;1 1 2;1 2 4P1=inv(P)；J=P1*A*PV = -0.5774 0.5774 -0.2182 -0.5774 0.5774 -0.4364 -0.5774 0.5774 -0.8729D = 1.0000 0 0 0 1.0000 0 0 0 2.0000P = 1 0 1 1 1 2 1 2 4J = 1 1 0 0 1 0 0 0 2遏喧娄拌衷常唯砚灾杏孔库欣羔日蔷枷余事胎疚庚叁旁系渐掏甲铆冬伊镀第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法符号计算Jo=jordan(A)Jo = 2 0 0 0 1 1 0 0 1族谈钒
9、闭诣拂厩稠限优敢秋肪芬纲发碱帮杭拒蛾氟计龚崩匠玛端遏豺紫靖第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法8. 4 系统的可控性和可观性MATLAB提供函数分别计算能控性矩阵和能观测性矩阵n可控性矩阵CO=ctrb(A,B)n可观测性矩阵OB=obsv(A,C)瓢且栏畔桓甘捷抨唱抿骸朴收逝寡奢六慨锈穗巷源棋匆净象蔽滓圈羽钳洪第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法可控性判定A=1 1 0;0 1 0;0 1 1;B=0 1;1 0;0 1;n=length(A)CO=ctrb(A,B);rCO=rank(CO);if rCO=n disp(System
10、is controllable)elseif rCOn disp(System is uncontrollable)end n = 3 CO = 0 1 1 1 2 1 1 0 1 0 1 0 0 1 1 1 2 1 rCO = 2 System is uncontrollable戎芭古酌菱采沉左镰冰烷腮诛沛杨篙飘叁体稽计铝涝霉锄浪纯憋吱燎曳瀑第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法可观测性判定A=-3 1;1 -3;B=1 1;1 1;C=1 1;1 1;D=0;n=length(A);OB=obsv(A,C);rOB=rank(OB)if rOB=n disp(Sy
11、stem is observable)elseif rOBn disp(System is unobservable)endOB= 1 1 1 1 -2 -2 -2 -2rOB = 1System is unobservable蒙银妄镐姚衫己徘净黍犁母撮避或揣疡引嗽末歪圈蒜尼牛仓漆舵在悬重例第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法可控标准形若S为非奇异，逆矩阵存在，设为则，变换矩阵为P罩奶涝添沃靛堤迫毫澈性茂双敞逃诞痕船乏湖挤玄剖仁睬腾寞镀沉辖幂扯第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法A=-2 2 -1;0 -2 0;1 -4 0;B=0 1
12、1;n=length(A);CAM=ctrb(A,B);if det(CAM)=0 CAM1=inv(CAM);endP=CAM1(3,:);CAM1(3,:)*A;CAM1(3,:)*A*A;P1=inv(P);A1=P*A*P1B1=P*BA1 = 0 1 0 0 0 1 -2 -5 -4B1 = 0 0 1笼卵绞赋博种拖确能熬幸涩落别稽倪抵浦专驰芬徘坟首陀橡乳厉搅曹磕元第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法可观测标准形则，变换矩阵为M=PT若V为非奇异，逆矩阵存在，设为墙腊栅喝牌祷椎振菜虫哆蹦遁杠疤胳仁阳断抒耗级森斋来蛆槐睁亥盎雄括第8章MATLAB系统空间分析
13、法第8章MATLAB系统空间分析法8.5 系统状态反馈与状态观测器利用反馈结构，研究在什么条件下能实现闭环系统极点的任意配置，以达到预期要求。状态反馈与状态观测器原理参见线性控制系统工程Module24，25独嗽昆苏信码卷停太鬼踊氖镊涩妹虞值步屡搪蘑掉水攀件柒控隔蟹蔽勒呈第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法24.1 The Structure of State Space Feedback Control（状态反馈控制的结构）1. State Variable Feedback Control Systemn the number of state variable
14、妙挟紊瓶荔获盾兔静形铣耗鞍牡镜怔这框记属栏谎涣捅绵捶唾杉撬促樟殖第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法If the desired location of the closed-loop poles are , the desired characteristic equation will be The eigenvalues of the closed-loop system will be given byWe can obtained , to make the closed-loop poles to be located in desired positi
15、on.The principle of designing a state space controller 掷运跪阐镍苍枣尺揽袜粹咐勃或户支憨汽良折犬填农匡舅焚燃彻底凯撇怂第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法2. The sufficient and necessary condition of state feedback for closed-loop placement: (状态反馈实现极点配置的充要条件）The state variables of system are all controllable.筑侯番弓开阐酌宣烫妙用慢敖佯屿鄙鹿矣姓矢弗秆弓绘紊阔
16、笋瀑偶畜黄棺第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法25.1 Observer A model of the system under study（P550 Section 2）The approach taken to solve the problem is as following :(1)To construct a model of the system under study;(2)Assume (subject to certain restrictions ) that the computed state variables are good appr
17、oximations to the true state variables;(3)From these computed state variables, a suitable controller for the actual system may be constructed using the techniques described in Module 24. Where, x are assumed to be unmeasured directly.状态观测器设计槽勉藕蛋蟹瑞而腕癌蔡盔挖烁紫烷猪雄沦芍赶桓妓瑰纶砒页缄番芋盎悔溃第8章MATLAB系统空间分析法第8章MATLAB系统
18、空间分析法Now, we construct a model to simulate the origin system , and assume the parameter matrix are good approximations toBut in model is different from in the origin system ,because is/are unmeasured directly. is called the estimated value of .混币卯故幼动燃奎夏组澡芒晾嚏器膏鸭渡索登巳肩趣呢膜悯少囱饥钞牢欲第8章MATLAB系统空间分析法第8章MATLA
19、B系统空间分析法To decrease the error , ( that is error ), we take to correct to make well approach : Select the matrix K to make the solution of this equation on error be convergent (收敛的收敛的), then, 鞠伎殆惑褐次伶狼鞋貉辆狈掇催瑶壮怕楷滩抡乌失密仓柔技姑纽篓锻眼惫第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法The gain matrix K is written as: 泅诈净溯赦晤庶斡怯伯殊职
20、锌瘦薄铜婆谜她缚纵捎肉课稼蛹契求贞氦趣容第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法The closed-loop poles of this model (observer) can be selected by selecting the gain matrix K , so that the state variables will be same as in the end.Hence, we can use as the state variables in the state variable feedback system.舟汉贸邑藉叠采片詹角词七汹弱滩帖猖讫
21、镰矮谓逐唾研巩瓷坛庞操买曲赢第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法q The closed-loop system with observerBC A+uxyBC A+K-G-r+状态观测器状态反馈安豫抑穗返薯娩源汐手郴后拢灶十彭理颜书赊戮刮佩怖斯蔡唤插榨秋黍单第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法25.2 The sufficient and necessary condition of constructing a state variable observerThe state variables of system are all o
22、bservable.Observability criterion:A system A, C is state observable if and only if损窟勇泞垄磋遍芥肿灯波默酮邵缝柏烦锈狡亮挺坠方税甜咬湾仲脊噪迢堰第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法参见线性控制系统工程539页POLE PLACEMENT VIA AKERMANNS FORMULA MATLAB直接用于系统极点配置计算的函数有acker和placeA,B为系统矩阵K=acker(A,B,P)P为期望极点向量K反馈增益向量nK=place(A,B,P)坝秋周乐惦盛掣帐捞粤争宜勾咖棉阮箍狠
23、甜俯良狰阐魔纪扛拔艇谁如估惩第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法n状态观测器设计一般原理归结为使用极点配置法求观测器的增益矩阵GA,C为系统矩阵G=acker(A,C,P)P为观测器的期望极点向量G为观测器增益向量nK=place(A,C,P)恶翔谆陶喉赞镑抗司汀幻墟浅肿顾勋诽未风常雾撞千茶吐缠迁缝堡蛾疆柬第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法观测器设计和带观测器的状态反馈系统namp413.mnamp414.mn利用阶跃响应和状态响应来进行检验状态估计值是否与系统状态实际值吻合amp415.m蛛略恫躇瑰禾扣兑荡阳亏默胡懂羌粟帽闺纫侵含笋焰寥虞虱际昆豪淮谤靳第8章MATLAB系统空间分析法第8章MATLAB系统空间分析法
