دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي استاد درس...

دانشگاه صنعتي اميركبيردانشكده مهندسي پزشكي

استاد درسدكتر فرزاد توحيدخواه

1389بهمن

بين-دکتر کنترل پيشتوحيدخواه

MPC Stability-1

Discrete-time MPC with Prescribed Degree of Stability



Finite Prediction Horizon: Re-visited

Example 4.1.


Condition number of the Hessian matrix increasesas the prediction horizon Np increases.


Origin of the Problem

Using Laguerre functions (for real time):


When there is an integrator in the system matrix A, the norms of the matrix power ||Am|| and the convolution sum ||φ(m)|| do not decay to zero, as m increases.Thus, the magnitudes of the elements in Ω increase as the prediction horizon Np increases. Hence, if the prediction horizon Np is large, a numerical conditioning problem occurs. This problem exists in the majority of the classical predictive controllers formulations, including GPC and DMC.


Traditional solution (previous chapter):

Use of an inner-loop state feedback stablization that may compromise the closed-loop performance when constraints become active, or the use of prediction horizon Np and control horizon Nc as the tuning parameters


Idea basis:

For a large Np, a large number is divided by another large number. This numerical problem becomes severe when the plant model itself is unstable, or when the dimension of the matrix A is large.


Solution:

1- Improving the numerical condition of MPC algorithms without guaranteeing closed-loop stability.

2- Asymptotic stability

3- Create a prescribed degree of closed-loop stability for the predictive control algorithm.


Use of Exponential Data Weighting


Continuous-time (in the LQR design):

eλt

Discrete-time:

{αj, j = 0, 1, 2 . . .},

α = eλt with t being the sampling interval.


Cost Function:

α = 1 the cost function becomes identical to the traditional cost function.


Exponentially Increasing Weight (α < 1):

Exponential weights α−2j , j = 1, 2, . . . ,Np, de-emphasizes the state x(ki + j | ki) at the current time and places emphasis on those at the future time.


Exponentially Decreasing Weight (α >1):

Exponential weights α−2j , j = 1, 2, . . . ,Np, more emphasizes the state x(ki + j | ki) at the current time and less emphasis on those at the future time.


Optimization of Exponentially Weighted Cost Function


Weighted incremental control:

Weighted state variable:


Theorem 4.1. The minimum solution of the exponentially weighted cost function J can be found by minimizing:


Example 4.2.

Consider the same double-integrator system given in Example 4.1. Examine how the parameter α used in the weighting affects the numerical condition and closed-loop control performance with constraints on the amplitude of the control signal as (only impose constraints on the first sample of the control)

α = 1/1.2 (exponentially increasing weight), α = 1 (no exponential weighting) and α = 1.2 (exponentially decreasing weighting)


1- with exponentially increasing weighting, the Hessian matrix is poorly conditioned even for short predictionhorizon;

2- without exponential weighting the condition number increases rapidly as the prediction horizon increases.

3- with exponentially decreasing data weighting, the condition number converges to a finite value and is much smaller than the one obtained without using exponential weighting.


Obviously, it is not feasible to use exponentially increasing weighting in this context, as the numerical condition rapidly deteriorates as prediction horizon increases, when α < 1.


Interpretation of Results from Exponential Weighting


The key point is that by transforming the exponentially weighted cost function to the traditional cost function, the augmented state-space model:

maximum modulusof all eigenvalues < 1

If


With this simple modification, intuitively we understand that there is no guarantee on the closed-loop stability with an arbitrary choice of α > 1.

However, when α is chosen to be slightly larger than one for the class of stable plants with embedded integrator, the closed-loop predictive systems are often found to be stable with Q = CTC and a diagonal R matrix with small positive elements.


For the first time, the prediction horizon Np can be selected to be sufficiently large to approximate the infinite prediction horizon case. Thus with Q ≥ 0 and R > 0, and sufficiently large (Np→∞), minimizing

is equivalent to the discrete-time linear quadratic regulator (DLQR) problem.


The traditional DLQR problem is solved using the algebraic Riccati equation

controllable observable


closed-loop system:

Because


if

closed-loop system is stable.


Second method:

For stability


By choosing α > 1, there is no guarantee that the closed-loop of the original system will be stable. But, if α is chosen to be slightly larger than unity, thenthe closed-loop system A−BK would often be stable.

Indeed, a large number of simulation tests show that this simple modification usually produces a stable closed-loop system, if the unstable modes from the augmented model come from the embedded integrators. However, a proper choice of the weightmatrices Q and R is important to create the degree of stability 1 − ε for the transformed system.


Asymptotic Closed-loop Stability with Exponential Weighting


Modification of Q and R Matrices

Basic idea:

The exponentially decreasing weight α > 1 increased the magnitudes of the actual closed-loop eigenvalues by the α factor. If the new Q and R matrices are selected to decrease the magnitudes of the eigen-values of the exponentially weighted system by a factor of α−1, then the magnitudes of the actual closed-loop eigenvalues become unchanged


Theorem 4.2.


Interpretation of the Results

The essence of the results lies in the fact that the two cost functions lead to the same optimal control. However, the commonly used cost function is limited to a finite prediction horizon for the class of predictive control algorithms that have embedded integrators. In contrast, the exponentially weighted cost function removes the problem because the model used in the prediction is modified to be stable using the factor α. As a result, the prediction horizon Np can be selected to be sufficiently large without numerical problems. Hence, asymptotic closed-loop stability is guaranteed


Example 4.3. Consider the simple double-integrator system described in 4.1

Design a MPC with an integrator for disturbance rejection,

Calculate the closed-loop eigenvalues, gain matrix via the cost function using exponential data weighting with α = 1.6 and compare the results with the case without weighting (α = 1)


With exponential data weighting


Without exponential data weighting (α = 1)


MIMO system


Example 4.4.

دانشگاه صنعتي اميركبيردانشكده مهندسي پزشكي

مبحث پايداریتنظيم

سجاد جعفرياستاد درس

دكتر فرزاد توحيدخواه1387بهمن


پايداری

MPC( خواص ny , nu با تغيير افق ها )متفاوت مي شود. يعني مثالً مي تواند حتي

پايدار و ناپايدار شود

خطي هنوز در حوزه پايداري و MPCحتي مقاوم بودن آن داراي مسائل جديد است

غيرخطي مسائل فوق حادتر شده و MPC در خطي MPCموضوعات جديدتري نسبت به

وجود دارد


GPCروشهای بررسی پايداري

سيستم 1 قطبهاي كردن )پايدار كالسPيك -روشهاي حلقPه بسPته )يPا مقاديPر ويژPه سPيستم حلقPه بسPته(( اگر قطب هPا يPا مقاديPر ويژPه در داخPل دايره واحPد بود سيستم پايدار اسPت. در غيPر ايPن صPورت سPيستم ناپايدار است.

(z=1)پايدار مرزي

بPا محدوديت MPC-حPل 2 هزينPه همراه تابPع بPا مقيPد حالت نهايي صفر


اشكاالت

(Hard Constraints)پيچيدگي روش ناشي از محدوديت سخت 1.

. خطاي آفست در خروجي2

(u. اشباع در ورودي )3

. امكان نرسيدن به پاسخ مطلوب4


GPCروشهای بررسی پايداري

- روش لياپانوف3

شبيه 4 روش -سازي


پايدار است اگر براي هر مربوط به :تعريف

:وجود داشته باشد به طوريكه مقدار

0X )(xfX 0

)(

ttXX ||)(|| ||)(||

روش لياپانوف



پايدار اگر اسPت ناپايدار .نباشد 0X مربوط به :تعريف )(xfX

0X مربوط به :تعريف )(xfX پايدار مجانبي است اگر پايدار باشد

را چنان يافت كهو بتوان ttxX )( ||)(||


قضيه لياپانوف


xfX)( يكي از نقاطP تعادل x=0فرض كنيد باشد. در اين صورت اگرVتابعي پايدار لياپانوفي است اگرx=0 باشد، آنگاه V(x)>0 پيوسته و مشتق پذير باشد و

) عالوه بر آن پايدار مجانبي است اگر ) 0V x ( ) 0V x


مثال


)()( kAxkx 1)( )()( PXxPkpxkxV TT

)()()()()()( kPxkxkPxkxkVkVV TT 111

QkxPPAAkxkPxkxkPAxAkxQ

TTTTT )()()()()()()(

Q.بايد منفي معين باشد


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