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程序代写案例-T3
时间：2021-11-23
2021 T3
ELEC 4632
Final checklist
By Sugar姐&Nathan
2021.10.22
Chapter 1
Naïve approach First step: we design a continuous-time controller to control this system.
Second step: we re-write the continuous-time control law as a system of differential equations.
Third step: we design a digital controller to approximate the continuous-time controller.
Time dependence The presence of sampling makes digital control system time-varying. Response depends on the time when
the step occurs. If the input is delayed, then the output is delayed by the same amount only if the delay is
multiple of the sampling period.
Deadbeat control The period nature of the control action can be actually used to obtain control strategies with superior
performance. The system is at rest when the desired position is reached. Only in discrete-time system, large
overshoot, shortest response time.在 n步之内达到稳定
Sampling creates new
frequency
There is a considerable variation of the measured signal over the sampling period and the low-frequency
variation is obtained by sampling the high-frequency signal at a slow rate.
Chapter 2
3 种 system
representation
Polynomial difference equation
State space equation
Transfer function
ODE-SS-TF 转
换三角图
ODE→SS: ( + 1) + 1( + − 1) + 2( + − 2) + ⋯ + −1( + 1) + () = ()
= [
0 1 … 0
0 0 1
… 1
− −−1 … −1
] , H = [
0
0
…
b
] , = [1 0 0 ⋯ 0]
SS→TF: () = ( − )−1 +
ODE→TF: Laplace transform后, () =
()
()
Type of sampling Periodic sampling, multiple-order sampling, multiple-rate sampling, random sampling
Sample devices Sampler: converts a continuous signal into a train of pulses occurring at the sampling instants 0, h, 2h…
Holding device: converts the sampled signal into one which is constant between two consecutive sampling
intervals. Transfer function is ℎ =
1−ⅇ−ℎ
Three method to
calculate
exponential
Numerical calculation
Series expansion of matrix exponential: ⅇℎ = ∑
(ℎ)
!
∞
=0
= + ℎ +
2ℎ2
2!
+ ⋯
ℎ
!
Laplace transform: (ℎ) = ⅇℎ = ℒ−1[( − )−1]
SS-TF，连续-离
散
四方图
Sample with
delay
0 ≤ ≤ ℎ > ℎ
( + 1)ℎ = (ℎ) + 0(ℎ) + 1(( − 1)ℎ) ( + 1)ℎ = (ℎ) + 0(ℎ − ( − 1)ℎ)
+ 1(ℎ − ℎ)
= ⅇℎ
0 = ∫ ⅇ
ℎ−
0
1 = ⅇ
(ℎ−) ∫ ⅇ
0
= ( − 1)ℎ + 0
= ⅇℎ
0 = ∫ ⅇ
ℎ−0
0
1 = ⅇ
(ℎ−0) ∫ ⅇ
0
0
() = ( − )−1(0 + 1
−1) +
Pulse transfer
function
() = (1 − −1) ∗ 变换 [ℒ−1(
()
)]
Holding device transfer function is ℎ =
1−ⅇ−ℎ
可能考框图题，见 tut4最后两题
Mason’s gain formula: 闭环系统=(输入到输出这条线上的 TF)/(1+每个圈圈上的 TF)
Chapter 3
Stability定义 Dynamic system with a bounded response to a bounded input
稳定域 连续系统：s域的左半平面
离散系统：z域以原点为圆心的单位圆内
直接判断 stability 开环稳定性
|ⅇ ⅇ| < 1
|ⅇ()| < 1
闭环稳定性
|ⅇ ⅇ| < 1
|ⅇ( − )| < 1
Routh定理
特殊情况：第一列为零，全零行，见超纲 pdf讲解
Jury定理 二阶系统：
() = 0
2 + 1 + 2
2 < 1, 2 > −1 + 1, 2 > −1 − 1 全满足的话，系统稳定（记得画倒三角图）
高阶系统：
Necessary condition
Sufficient condition
() = 0
+ 1
−1 + ⋯ + −1 +
稳定条件：对于() = 0
+ 1
−1 + ⋯ + −1 + 满足
A(1)>0
(−1)(−1) > 0
0 > ||, 0 > |−1|, 0 > |−2|,…
Hurwitz 定理 矩阵所有左上角的子矩阵全都 PD，则矩阵 PD，详解询糖
四个 ability 的关系 定义：
Reachability: any final state can be reached from any initial state.
Controllability: the origin can be reached from any initial state.
Unobservable: there exist a finite time that y(k)=0 and u(k)=0.
Observability: inputs and outputs are sufficient to determine the initial state of the system.
Detectability: all unobservable states are decay to the origin.
关系：
Reachability → Controllability, Controllability 不→ Reachability
Observability → Detectability, Detectability 不→ Observability
公式：
Reachability: = [
2 ⋯ −1] full rank
Observability: 0 = [ CG
2 ⋯ −1] full rank
特例：
Controllable but not reachable:
= [
0 0
1 0
]， = [0
1
]
Detectable but not observable:
= [
1.1 −0.3
1 0
]， = [0
1
]， = [1，− 0.5]
相关题目：
1. Reachability一定可以推出controllability，因为定义（抄一遍），controllability包含在reachability
里
2. Controllability不一定推出reachability，反例（这个系统controllable但是不reachable）
= [
0 0
1 0
]， = [0
1
]
系统(+1)=()+() 是controllable的，因为2=0，那么当第0时刻和第一时刻的输入
(0)=(1)=0的时候，两步之内origin can be reached from any initial state。
(1)=(0)+(0)=(0)
(2)=(1)+(1)= 2(0)=0
达到origin了，就是controllable
但是reachability matrix =[
0 0
1 0
] 不是full rank，所以不是reachable的
相关题目：
3. Observability一定可以推出detectability，因为定义（抄一遍），detectability包含在observability里
4. Detectability不一定推出observability，反例（这个系统detectable但是不observable）
= [
1.1 −0.3
1 0
]， = [0
1
]， = [1，− 0.5]
observability matrix =[
1 −0.5
0.6 −0.3
] 不是full rank，所以不是observable的
如图，所有 states最后 decay to the origin了，所以是 detectable的
Sampling loss ability To get a reachable discrete-time system, it is necessary that the continuous-time system is reachable.
It may happen that reachability or observability is lost for some sampling periods.
Chapter 4
Pole placement design
目的
Find linear feedback to form closed loop system
Ackermann’s formula () = + 1
−1 + 2
−2 … −1 + = 0 (此处 P为系数 并非 poles)
() = + 1
−1 + 2
−2 … −1 + = 0
Controller gain L: = [0 ⋯ 01] ×
−1 × ()
Observer gain K: = () × 0
−1 × [0 ⋯ 01]
Deadbeat: P(G) =
设计 controller和 observer之前一定要检测Wc和Wo是不是 full rank
求 controller gain L和
observer gain K
配平法
ⅇ[ − ( − )] = ( − ⅇ1)( − ⅇ2) … ( − ⅇ)
ⅇ[ − ( − )] = ( − ⅇ1)( − ⅇ2) … ( − ⅇ)
此处 poles为 closed loop poles
Deadbeat:
ⅇ[ − ( − )] =
ⅇ[ − ( − )] =
State estimation
过程
Real system: ( + 1) = () + () ①
Estimated system: ̂( + 1|) = ̂(| − 1) + () + [() − ̂()] ②
Let error: ̃ = − ̂
①-② Real system-estimated system:
̃( + 1|) = ̃(| − 1) + [(| − 1) − ̂(| − 1)]
= ̃(| − 1) − ̃(| − 1)
= ( − )̃(| − 1)
Design observer to find K using Ackermann’s formula to let
| − | stable, that is |ⅇ( − )|<1
state estimation error ̃ → 0
Output feedback-
feedback controller过
程
Real system: ( + 1) = () + () ①
Controller gain: u() = −̂() ②
Estimated system: ̂( + 1|) = ̂(| − 1) + () + [() − ̂()] ③
Let error: ̃ = − ̂
①-③ Real system-estimated system:
̃( + 1|) = ( − )̃(| − 1)
Substitute ② into ①
( + 1) = () + ()
= () + (−̂())
= () + ∗ (̃(| − 1) − ())
= ( − ) ∗ () + ∗ ̃(| − 1)
By augmentation
[
̃( + 1|)
()
] = [
( − ) 0
( − )
] ∗ [
̃(| − 1)
()
]
= [
( − ) 0
( − )
]
Make ⅇ() inside unit disc, is to find L and K for output feedback controller
That is, design L and K to make ( − ) stable and ( − ) stable
Chapter 5
A good control
system have 5
properties
1. It should be stable
2. It should operate with as little error as possible
3. The system should exhibit suitable damping
4. The system performance should not be appreciably affected by small changes in certain parameters
(robustness)
5. The system should be able to mitigate the effect of undesirable disturbances
Steady state error Response 有两个时间段
Transient: from initial to final state
Steady state: time goes to infinity
Final value theorem
ⅇ =
→∞
(ⅇ) =
→1
(1 − −1)()
查表方法： system type: number of poles at z=1
Input Type 0 Type 1 Type 2
Step
() =
() =
− 1
ⅇ =
1 +
0 0
Ramp
() =
() =
ℎ
( − 1)2
∞ ⅇ =
0
Parabolic
() =
1
2
2
() =
ℎ2( + 1)
2( − 1)3
∞ ∞ ⅇ =
Open loop transfer function ()
=
→1
()
=
→1
(1 − −1)()
=
→1
(1 − −1)2()
Closed loop steady state error is ⅇ
Robust Feedback可以减小 sensitivity of system to parameter variation
Ess+robust:
Final value theorem
ⅇ =
→∞
(ⅇ) =
→1
(1 − −1)()
Disturbance
rejection
Disturbance: unwanted input signal that affect output
Ess+ disturbance:
Final value theorem
ⅇ =
→∞
(ⅇ) =
→1
(1 − −1)()
Suitable damping
Transient response
described in 2
factors
1. Swiftness: as represented by the rise time and the peak time
2. Closeness:represented by the overshoot and settling time
Overshoot: the amount the system output response proceeds beyond the desired response
Settling time: the time required for the system output to settle within a certain percentage of the input amplitude
Chapter 6
非线性系统的特点 1. Dependence of the system response behavior upon the magnitude and type of the input (input 的
magnitude 和 type不同，response不同)
2.
Nonlinear system we use Time-invariant system: ( + 1) = ((), ())
Autonomous system: () ≡ 0
( + 1) = (())
Equilibrium point
(又叫 Singular point)
ⅇ = (ⅇ)
判断 EP是不是唯一的：
If matrix ( − ) not singular, ⅇ = 0 is the isolated equilibrium point, no other EPs around it
If matrix ( − ) singular, 有很多 EP ⅇ, not isolated
Singular matrix: 奇异矩阵，det=0
Stability analysis of
singular points
For system ( + 1) = (())
Suppose = [1, 2, ⋯ ], let ̃ = [̃1, ̃2, ⋯ , ̃] be a singular point
Let us now consider a ball of finite radius > 0 surrounding the point 0 of the state space
The set of points described by the inequality (1 − ̃1)
2 + (2 − ̃2)
2 + ⋯ +( − ̃)
2 ≤ 2
Let this region be denoted by (̃, )
A singular point ̃ of the system is said to be stable in the sense of Lyapunov if there exists a region
(̃, ) such that any solution starting from the point 0 in this region does not go outside the region
(̃, )
For system ( + 1) = (())
Suppose = [1, 2, ⋯ ], let ̃ = [̃1, ̃2, ⋯ , ̃] be a singular point
Let us now consider a ball of finite radius > 0 surrounding the point 0 of the state space
The set of points described by the inequality (1 − ̃1)
2 + (2 − ̃2)
2 + ⋯ +( − ̃)
2 ≤ 2
Let this region be denoted by (̃, )
A singular point ̃ of the system is said to be asymptotically stable if there exists a region (̃, ) such
that any solution starting from the point 0 in this region does not go outside the region (̃, ) and
() → ̃ as → ∞
Only isolated singular points can be asymptotically stable
For system ( + 1) = (())
Suppose = [1, 2, ⋯ ], let ̃ = [̃1, ̃2, ⋯ , ̃] be a singular point
Let us now consider a ball of finite radius > 0 surrounding the point 0 of the state space
The set of points described by the inequality (1 − ̃1)
2 + (2 − ̃2)
2 + ⋯ +( − ̃)
2 ≤ 2
Let this region be denoted by (̃, )
A singular point ̃ of the system is said to be stable in the sense of Lyapunov if there exists a region
(̃, ) such that any solution starting from the point 0 in this region does not go outside the region
(̃, )
A singular point ̃ of the system is said to be globally asymptotically stable if it is stable in the sense of
Lyapunov and () → ̃ as → ∞ for any solution () to the system
It is obvious that if a singular point is globally asymptotically stable, then this point is the only singular
point of the system.
In linear system: asymptotically stable = globally asymptotically stable
In nonlinear system: asymptotically stable ≠ globally asymptotically stable
Linearization For system ( + 1) = (()) do Tayler serios expansion
() = (̃) +
( − ̃) +
1
2!
2
2
( − ̃)2 …
Neglect higher order terms, (̃) = 0, the equation may be linearized as
( + 1) = ()
=
|
=
The behavior of the nonlinear system in a small region surrounding the ̃ may be approximated by the
linear system.
ⅇ() Linear system stability Original nonlinear system EP stability
|ⅇ()| < 1 Asymptotically stable Asymptotically stable
|ⅇ()| > 1 Unstable Unstable
|ⅇ()| = 1 Unknown Unknown
Lyapunov stability For system ( + 1) = (())
Suppose there exists a scalar Lyapunov function V() continuous in x such that
1. () > 0 ≠ 0
2. () < 0 ≠ 0,
(()) = (( + 1)) − (()) = ((())) − (())
3. (0) = 0
4. () → ∞ || → ∞
Then the equilibrium state = 0 is globally asymptotically stable
Lyapunov stability for
linear system
设计 Lyapunov function
For system ( + 1) = ()
Choose possible Lyapunov function (()) = ()()
Where P is a positive definite symmetric matrix
(()) = (( + 1)) − (()) = ( + 1)( + 1) − ()()
= [()][()] − ()() = ()() − ()()
= ()( − )()
Let = −( − ) positive definite
Specify a positive definite real symmetric matrix Q then see whether or not matrix P is positive definite,
positive definite P is necessary and sufficient condition
Effect of discretization on
stability
连续系统离散系统稳定
域区别
Continuous time system Discrete time system
̇ = ( + 1) = ()
= ⅇℎ , ℎ > 0
= ⅇℎ > 1 > 0
= ⅇℎ = 1 = 0
= ⅇℎ < 1 < 0
So stable range is mapped from continuous time into discrete time
Chapter 8
Optimal control advantages 1. They easily handle MIMO systems
2. They are relatively easy to understand
3. They lead to computationally efficient algorithms
4. There is a huge number of industrial applications, especially, applications of Model Predictive
Control
General optimality principle General optimal control problem
Find an optimal input 0() for = 0,1, ⋯ so that the cost function ≔ ∑ ((), (), )
=0
is minimized for the system ( + 1) = ((), (), ) with initial condition (0) = 0 . This
problem can be solved using Dynamic Programming.
Optimality principle,
Bellman principle, Dynamic
programming principle
If an input 0() for = 0,1, ⋯ is the optimal solution for the above problem, the
0() is also
the optimal solution over any subinterval = 0, 0 + 1, … , where 0 < 0 <
Linear quadratic optimal
control (LQR)
For system ( + 1) = () + () with initial condition (0) = 0
Consider quadratic cost function ≔ ∑ [()() + ()()]
=0
Q, R are symmetric weighting matrices, Q is nonnegative definite, R is positive definite
Linear regulator () = −()(), where () is defined by the equations
() = 0
() = [( + 1) − ( + 1)−1( + 1)] + Discrete Time Dynamic Riccati Equation
() = [ + ( + 1)−1]( + 1)
Optimal controller is linear and time-varying, controller gain () does not depend on initial
condition
Steady state linear quadratic
optimal control
(infinity time)
≔ ∑ [()() + ()()]
∞
=0
is infinity cost function
() → ∞ time-invariant for all k as → ∞
= [ − −1] + Discrete Time Algebraic Riccati Equation
∞ = [ +
−1]
Model Predictive Control
(MPC)
MPC is used in many industrial fields
Algorithm: Based on solving an on-line problem of optimization with constraints.
1. At time k for current state (), solve, on-line, and open-loop optimal control problem with
constraints
2. Apply the first step in the optimal control sequence
3. Repeat the procedure at time (k+1), using the current state ( + 1)
Give an example 题 1根 stable 3根 unstable
1( + 1) = 1
2() − 0.51()
2( + 1) = −32
2()
1根 stable 1根 unstable
1( + 1) = 2
2()
2( + 1) = 1()
2根 stable 2根 unstable
1( + 1) = 1
4() − 0.51
2() + 1.51()
2( + 1) = −0.52()
1根 unstable的三阶系统
( + 1) = ()
( + 1) = −0.5() − 2()
( + 1) = −0.5() − 2()
1根 unstable的二阶系统
( + 1) = ()
( + 1) = 0.5() + 2()
ELEC 4632
Final checklist
By Sugar姐&Nathan
2021.10.22
Chapter 1
Naïve approach First step: we design a continuous-time controller to control this system.
Second step: we re-write the continuous-time control law as a system of differential equations.
Third step: we design a digital controller to approximate the continuous-time controller.
Time dependence The presence of sampling makes digital control system time-varying. Response depends on the time when
the step occurs. If the input is delayed, then the output is delayed by the same amount only if the delay is
multiple of the sampling period.
Deadbeat control The period nature of the control action can be actually used to obtain control strategies with superior
performance. The system is at rest when the desired position is reached. Only in discrete-time system, large
overshoot, shortest response time.在 n步之内达到稳定
Sampling creates new
frequency
There is a considerable variation of the measured signal over the sampling period and the low-frequency
variation is obtained by sampling the high-frequency signal at a slow rate.
Chapter 2
3 种 system
representation
Polynomial difference equation
State space equation
Transfer function
ODE-SS-TF 转
换三角图
ODE→SS: ( + 1) + 1( + − 1) + 2( + − 2) + ⋯ + −1( + 1) + () = ()
= [
0 1 … 0
0 0 1
… 1
− −−1 … −1
] , H = [
0
0
…
b
] , = [1 0 0 ⋯ 0]
SS→TF: () = ( − )−1 +
ODE→TF: Laplace transform后, () =
()
()
Type of sampling Periodic sampling, multiple-order sampling, multiple-rate sampling, random sampling
Sample devices Sampler: converts a continuous signal into a train of pulses occurring at the sampling instants 0, h, 2h…
Holding device: converts the sampled signal into one which is constant between two consecutive sampling
intervals. Transfer function is ℎ =
1−ⅇ−ℎ
Three method to
calculate
exponential
Numerical calculation
Series expansion of matrix exponential: ⅇℎ = ∑
(ℎ)
!
∞
=0
= + ℎ +
2ℎ2
2!
+ ⋯
ℎ
!
Laplace transform: (ℎ) = ⅇℎ = ℒ−1[( − )−1]
SS-TF，连续-离
散
四方图
Sample with
delay
0 ≤ ≤ ℎ > ℎ
( + 1)ℎ = (ℎ) + 0(ℎ) + 1(( − 1)ℎ) ( + 1)ℎ = (ℎ) + 0(ℎ − ( − 1)ℎ)
+ 1(ℎ − ℎ)
= ⅇℎ
0 = ∫ ⅇ
ℎ−
0
1 = ⅇ
(ℎ−) ∫ ⅇ
0
= ( − 1)ℎ + 0
= ⅇℎ
0 = ∫ ⅇ
ℎ−0
0
1 = ⅇ
(ℎ−0) ∫ ⅇ
0
0
() = ( − )−1(0 + 1
−1) +
Pulse transfer
function
() = (1 − −1) ∗ 变换 [ℒ−1(
()
)]
Holding device transfer function is ℎ =
1−ⅇ−ℎ
可能考框图题，见 tut4最后两题
Mason’s gain formula: 闭环系统=(输入到输出这条线上的 TF)/(1+每个圈圈上的 TF)
Chapter 3
Stability定义 Dynamic system with a bounded response to a bounded input
稳定域 连续系统：s域的左半平面
离散系统：z域以原点为圆心的单位圆内
直接判断 stability 开环稳定性
|ⅇ ⅇ| < 1
|ⅇ()| < 1
闭环稳定性
|ⅇ ⅇ| < 1
|ⅇ( − )| < 1
Routh定理
特殊情况：第一列为零，全零行，见超纲 pdf讲解
Jury定理 二阶系统：
() = 0
2 + 1 + 2
2 < 1, 2 > −1 + 1, 2 > −1 − 1 全满足的话，系统稳定（记得画倒三角图）
高阶系统：
Necessary condition
Sufficient condition
() = 0
+ 1
−1 + ⋯ + −1 +
稳定条件：对于() = 0
+ 1
−1 + ⋯ + −1 + 满足
A(1)>0
(−1)(−1) > 0
0 > ||, 0 > |−1|, 0 > |−2|,…
Hurwitz 定理 矩阵所有左上角的子矩阵全都 PD，则矩阵 PD，详解询糖
四个 ability 的关系 定义：
Reachability: any final state can be reached from any initial state.
Controllability: the origin can be reached from any initial state.
Unobservable: there exist a finite time that y(k)=0 and u(k)=0.
Observability: inputs and outputs are sufficient to determine the initial state of the system.
Detectability: all unobservable states are decay to the origin.
关系：
Reachability → Controllability, Controllability 不→ Reachability
Observability → Detectability, Detectability 不→ Observability
公式：
Reachability: = [
2 ⋯ −1] full rank
Observability: 0 = [ CG
2 ⋯ −1] full rank
特例：
Controllable but not reachable:
= [
0 0
1 0
]， = [0
1
]
Detectable but not observable:
= [
1.1 −0.3
1 0
]， = [0
1
]， = [1，− 0.5]
相关题目：
1. Reachability一定可以推出controllability，因为定义（抄一遍），controllability包含在reachability
里
2. Controllability不一定推出reachability，反例（这个系统controllable但是不reachable）
= [
0 0
1 0
]， = [0
1
]
系统(+1)=()+() 是controllable的，因为2=0，那么当第0时刻和第一时刻的输入
(0)=(1)=0的时候，两步之内origin can be reached from any initial state。
(1)=(0)+(0)=(0)
(2)=(1)+(1)= 2(0)=0
达到origin了，就是controllable
但是reachability matrix =[
0 0
1 0
] 不是full rank，所以不是reachable的
相关题目：
3. Observability一定可以推出detectability，因为定义（抄一遍），detectability包含在observability里
4. Detectability不一定推出observability，反例（这个系统detectable但是不observable）
= [
1.1 −0.3
1 0
]， = [0
1
]， = [1，− 0.5]
observability matrix =[
1 −0.5
0.6 −0.3
] 不是full rank，所以不是observable的
如图，所有 states最后 decay to the origin了，所以是 detectable的
Sampling loss ability To get a reachable discrete-time system, it is necessary that the continuous-time system is reachable.
It may happen that reachability or observability is lost for some sampling periods.
Chapter 4
Pole placement design
目的
Find linear feedback to form closed loop system
Ackermann’s formula () = + 1
−1 + 2
−2 … −1 + = 0 (此处 P为系数 并非 poles)
() = + 1
−1 + 2
−2 … −1 + = 0
Controller gain L: = [0 ⋯ 01] ×
−1 × ()
Observer gain K: = () × 0
−1 × [0 ⋯ 01]
Deadbeat: P(G) =
设计 controller和 observer之前一定要检测Wc和Wo是不是 full rank
求 controller gain L和
observer gain K
配平法
ⅇ[ − ( − )] = ( − ⅇ1)( − ⅇ2) … ( − ⅇ)
ⅇ[ − ( − )] = ( − ⅇ1)( − ⅇ2) … ( − ⅇ)
此处 poles为 closed loop poles
Deadbeat:
ⅇ[ − ( − )] =
ⅇ[ − ( − )] =
State estimation
过程
Real system: ( + 1) = () + () ①
Estimated system: ̂( + 1|) = ̂(| − 1) + () + [() − ̂()] ②
Let error: ̃ = − ̂
①-② Real system-estimated system:
̃( + 1|) = ̃(| − 1) + [(| − 1) − ̂(| − 1)]
= ̃(| − 1) − ̃(| − 1)
= ( − )̃(| − 1)
Design observer to find K using Ackermann’s formula to let
| − | stable, that is |ⅇ( − )|<1
state estimation error ̃ → 0
Output feedback-
feedback controller过
程
Real system: ( + 1) = () + () ①
Controller gain: u() = −̂() ②
Estimated system: ̂( + 1|) = ̂(| − 1) + () + [() − ̂()] ③
Let error: ̃ = − ̂
①-③ Real system-estimated system:
̃( + 1|) = ( − )̃(| − 1)
Substitute ② into ①
( + 1) = () + ()
= () + (−̂())
= () + ∗ (̃(| − 1) − ())
= ( − ) ∗ () + ∗ ̃(| − 1)
By augmentation
[
̃( + 1|)
()
] = [
( − ) 0
( − )
] ∗ [
̃(| − 1)
()
]
= [
( − ) 0
( − )
]
Make ⅇ() inside unit disc, is to find L and K for output feedback controller
That is, design L and K to make ( − ) stable and ( − ) stable
Chapter 5
A good control
system have 5
properties
1. It should be stable
2. It should operate with as little error as possible
3. The system should exhibit suitable damping
4. The system performance should not be appreciably affected by small changes in certain parameters
(robustness)
5. The system should be able to mitigate the effect of undesirable disturbances
Steady state error Response 有两个时间段
Transient: from initial to final state
Steady state: time goes to infinity
Final value theorem
ⅇ =
→∞
(ⅇ) =
→1
(1 − −1)()
查表方法： system type: number of poles at z=1
Input Type 0 Type 1 Type 2
Step
() =
() =
− 1
ⅇ =
1 +
0 0
Ramp
() =
() =
ℎ
( − 1)2
∞ ⅇ =
0
Parabolic
() =
1
2
2
() =
ℎ2( + 1)
2( − 1)3
∞ ∞ ⅇ =
Open loop transfer function ()
=
→1
()
=
→1
(1 − −1)()
=
→1
(1 − −1)2()
Closed loop steady state error is ⅇ
Robust Feedback可以减小 sensitivity of system to parameter variation
Ess+robust:
Final value theorem
ⅇ =
→∞
(ⅇ) =
→1
(1 − −1)()
Disturbance
rejection
Disturbance: unwanted input signal that affect output
Ess+ disturbance:
Final value theorem
ⅇ =
→∞
(ⅇ) =
→1
(1 − −1)()
Suitable damping
Transient response
described in 2
factors
1. Swiftness: as represented by the rise time and the peak time
2. Closeness:represented by the overshoot and settling time
Overshoot: the amount the system output response proceeds beyond the desired response
Settling time: the time required for the system output to settle within a certain percentage of the input amplitude
Chapter 6
非线性系统的特点 1. Dependence of the system response behavior upon the magnitude and type of the input (input 的
magnitude 和 type不同，response不同)
2.
Nonlinear system we use Time-invariant system: ( + 1) = ((), ())
Autonomous system: () ≡ 0
( + 1) = (())
Equilibrium point
(又叫 Singular point)
ⅇ = (ⅇ)
判断 EP是不是唯一的：
If matrix ( − ) not singular, ⅇ = 0 is the isolated equilibrium point, no other EPs around it
If matrix ( − ) singular, 有很多 EP ⅇ, not isolated
Singular matrix: 奇异矩阵，det=0
Stability analysis of
singular points
For system ( + 1) = (())
Suppose = [1, 2, ⋯ ], let ̃ = [̃1, ̃2, ⋯ , ̃] be a singular point
Let us now consider a ball of finite radius > 0 surrounding the point 0 of the state space
The set of points described by the inequality (1 − ̃1)
2 + (2 − ̃2)
2 + ⋯ +( − ̃)
2 ≤ 2
Let this region be denoted by (̃, )
A singular point ̃ of the system is said to be stable in the sense of Lyapunov if there exists a region
(̃, ) such that any solution starting from the point 0 in this region does not go outside the region
(̃, )
For system ( + 1) = (())
Suppose = [1, 2, ⋯ ], let ̃ = [̃1, ̃2, ⋯ , ̃] be a singular point
Let us now consider a ball of finite radius > 0 surrounding the point 0 of the state space
The set of points described by the inequality (1 − ̃1)
2 + (2 − ̃2)
2 + ⋯ +( − ̃)
2 ≤ 2
Let this region be denoted by (̃, )
A singular point ̃ of the system is said to be asymptotically stable if there exists a region (̃, ) such
that any solution starting from the point 0 in this region does not go outside the region (̃, ) and
() → ̃ as → ∞
Only isolated singular points can be asymptotically stable
For system ( + 1) = (())
Suppose = [1, 2, ⋯ ], let ̃ = [̃1, ̃2, ⋯ , ̃] be a singular point
Let us now consider a ball of finite radius > 0 surrounding the point 0 of the state space
The set of points described by the inequality (1 − ̃1)
2 + (2 − ̃2)
2 + ⋯ +( − ̃)
2 ≤ 2
Let this region be denoted by (̃, )
A singular point ̃ of the system is said to be stable in the sense of Lyapunov if there exists a region
(̃, ) such that any solution starting from the point 0 in this region does not go outside the region
(̃, )
A singular point ̃ of the system is said to be globally asymptotically stable if it is stable in the sense of
Lyapunov and () → ̃ as → ∞ for any solution () to the system
It is obvious that if a singular point is globally asymptotically stable, then this point is the only singular
point of the system.
In linear system: asymptotically stable = globally asymptotically stable
In nonlinear system: asymptotically stable ≠ globally asymptotically stable
Linearization For system ( + 1) = (()) do Tayler serios expansion
() = (̃) +
( − ̃) +
1
2!
2
2
( − ̃)2 …
Neglect higher order terms, (̃) = 0, the equation may be linearized as
( + 1) = ()
=
|
=
The behavior of the nonlinear system in a small region surrounding the ̃ may be approximated by the
linear system.
ⅇ() Linear system stability Original nonlinear system EP stability
|ⅇ()| < 1 Asymptotically stable Asymptotically stable
|ⅇ()| > 1 Unstable Unstable
|ⅇ()| = 1 Unknown Unknown
Lyapunov stability For system ( + 1) = (())
Suppose there exists a scalar Lyapunov function V() continuous in x such that
1. () > 0 ≠ 0
2. () < 0 ≠ 0,
(()) = (( + 1)) − (()) = ((())) − (())
3. (0) = 0
4. () → ∞ || → ∞
Then the equilibrium state = 0 is globally asymptotically stable
Lyapunov stability for
linear system
设计 Lyapunov function
For system ( + 1) = ()
Choose possible Lyapunov function (()) = ()()
Where P is a positive definite symmetric matrix
(()) = (( + 1)) − (()) = ( + 1)( + 1) − ()()
= [()][()] − ()() = ()() − ()()
= ()( − )()
Let = −( − ) positive definite
Specify a positive definite real symmetric matrix Q then see whether or not matrix P is positive definite,
positive definite P is necessary and sufficient condition
Effect of discretization on
stability
连续系统离散系统稳定
域区别
Continuous time system Discrete time system
̇ = ( + 1) = ()
= ⅇℎ , ℎ > 0
= ⅇℎ > 1 > 0
= ⅇℎ = 1 = 0
= ⅇℎ < 1 < 0
So stable range is mapped from continuous time into discrete time
Chapter 8
Optimal control advantages 1. They easily handle MIMO systems
2. They are relatively easy to understand
3. They lead to computationally efficient algorithms
4. There is a huge number of industrial applications, especially, applications of Model Predictive
Control
General optimality principle General optimal control problem
Find an optimal input 0() for = 0,1, ⋯ so that the cost function ≔ ∑ ((), (), )
=0
is minimized for the system ( + 1) = ((), (), ) with initial condition (0) = 0 . This
problem can be solved using Dynamic Programming.
Optimality principle,
Bellman principle, Dynamic
programming principle
If an input 0() for = 0,1, ⋯ is the optimal solution for the above problem, the
0() is also
the optimal solution over any subinterval = 0, 0 + 1, … , where 0 < 0 <
Linear quadratic optimal
control (LQR)
For system ( + 1) = () + () with initial condition (0) = 0
Consider quadratic cost function ≔ ∑ [()() + ()()]
=0
Q, R are symmetric weighting matrices, Q is nonnegative definite, R is positive definite
Linear regulator () = −()(), where () is defined by the equations
() = 0
() = [( + 1) − ( + 1)−1( + 1)] + Discrete Time Dynamic Riccati Equation
() = [ + ( + 1)−1]( + 1)
Optimal controller is linear and time-varying, controller gain () does not depend on initial
condition
Steady state linear quadratic
optimal control
(infinity time)
≔ ∑ [()() + ()()]
∞
=0
is infinity cost function
() → ∞ time-invariant for all k as → ∞
= [ − −1] + Discrete Time Algebraic Riccati Equation
∞ = [ +
−1]
Model Predictive Control
(MPC)
MPC is used in many industrial fields
Algorithm: Based on solving an on-line problem of optimization with constraints.
1. At time k for current state (), solve, on-line, and open-loop optimal control problem with
constraints
2. Apply the first step in the optimal control sequence
3. Repeat the procedure at time (k+1), using the current state ( + 1)
Give an example 题 1根 stable 3根 unstable
1( + 1) = 1
2() − 0.51()
2( + 1) = −32
2()
1根 stable 1根 unstable
1( + 1) = 2
2()
2( + 1) = 1()
2根 stable 2根 unstable
1( + 1) = 1
4() − 0.51
2() + 1.51()
2( + 1) = −0.52()
1根 unstable的三阶系统
( + 1) = ()
( + 1) = −0.5() − 2()
( + 1) = −0.5() − 2()
1根 unstable的二阶系统
( + 1) = ()
( + 1) = 0.5() + 2()
